Why is it apparently not dangerous to fire a shotgun (such as when "skeet shooting") into the air? If you fire a gun or rifle into the air, whether straight up or at an angle, as I understand physics, a metal projectile will gain surprising momentum on its way down again, more than capable of killing a grown human being, not to mention small animals and children, property, etc.
I have never fired a gun, but I assume that one of the first things they tell you is to never point the gun at a living being unless you intend to kill them, and next after that, to never fire up in the sky.
However, people "skeet shoot" all the time, and seem to not apply this basic safety measure whatsoever when it comes to shotguns. I understand that shotguns work in a different manner from a normal "bullet", instead causing tons of small particles to spread out, but still, won't those small particles also come back to Earth in the same manner as the lethal metal bullet?
 A: Mythbusters did a segment on this back in the day.
If you fire a rifle or pistol straight in the sky, at the top of its climb, the bullet loses the spin imparted by the rifling in the barrel and becomes a tumbling object falling from the sky. Even something as "heavy" as a rifle or pistol slug, when tumbling un-stabilized, is almost always non-lethal.
HOWEVER: firing exactly straight up into the sky is almost impossible for a human. If the trajectory of the bullet is not straight up and straight back down, it will retain the spin imparted by the rifling, thus regaining much more velocity on the way back down than if it was tumbling "loose". There are recorded instances of rifle and pistol slugs that retained an arc trajectory indeed being lethal. As has been said, Rule Number One of firearms is to never shoot if you don't know that where the bullet is going is something you intend to destroy.
Shotgun shells that you use to shoot skeet fire much smaller pellets, and there is no way to impart a spin to stabilize the trajectory and velocity of the individual pellets. Additionally, each pellet is much smaller and lighter than a rifle or pistol slug, and therefore the mass-to-surface area ratio is much smaller.
Shotguns with skeet shells are non-lethal to humans beyond like 50 meters or so, fired in a more or less direct trajectory. If you shot a skeet shell straight up in the air, any pellets striking you on the way down wouldn't hit any harder than comparably-sized hail.
A: My answer is from real world experience, not mathematical or physics. I’ve been dove hunting with others. This requires the same shot pellet size as skeet shooting. It was extremely common for a hunter to shoot at a dove and the pellets to fall back to the earth right on top of me. The pellets felt like rain drops and did not sting, hurt or leave a mark. The key is I was far enough down range that the pellets had arced up and lost their energy. As they were coming back down their terminal velocity was slow enough to not cause harm. Larger pellets like 00 buck or a slug would still do damage though.
A: One of the other rules of firearm safety is "Know your target and what is beyond it."
A firearm (of any sort) can be safely fired if there is nothing of value between the muzzle and the point where the projectile loses the last of its kinetic energy.  Some of these safe places are called "firing ranges". 
"Firearms, The Law, and Forensic Ballistics" by Margaret-Ann Armour provides formulae for the maximum range of a shotgun pellet based on the pellet diameter ($PD$):
$$r_\mathrm{yards} = 2200 \tfrac{\mathrm{yd}}{\mathrm{in.}} \times PD_\mathrm{inches}$$
or
$$r_\mathrm{meters} = 100 \tfrac{\mathrm{m}}{\mathrm{mm}} \times PD_\mathrm{millimeters}$$
When shooting skeet as in your question, you might select a shot between .110 inch (2.79 mm) and .080 inch (2.03 mm).  Doing the math, we come up with approximately 175-275 meters total projectile distance. That range (of values) is the maximum range (in meters) you need to make a (firing) range...and most shots will fall harmlessly to Earth much, much closer than the maximum. 

You can see this when you look at a satellite image of a shotgun range. There is nothing of value within 300 meters of the firing line, but most of the shot falls to Earth within 50m.
Oh, and just to drive home the point that we are really, really certain there won't be any risk beyond those distances, what could that be in the top-left corner of the image?

It's an airport!
A: In various parts of the world, firing into the air is a very standard gesture of high emotions at celebrations or at funerals.  The West maintains this as a traditional part of military funerals.
At military funerals, however, the guns are loaded with blanks and this is purely for display, because we don't want to create more funerals!  Elsewhere though, this does kill people.  The link above provides a long list of cases; most notable in the US are probably Armando Martinez, a member of the Texas House of Representatives, and Shannon Smith, whose death prompted Arizona to make this illegal.
So why don't shotgun pellets injure people?  Simple answer: they do.  As Dick Cheney knows, accidents happen when you aren't very careful about where you're firing and who's over there!
This gives you the answer why skeet shooting doesn't injure people with falling shot.  The location and direction of fire are carefully chosen so that nothing important is in the arc of fire, within the range of a shotgun pellet.  If this is in an area which is likely to have members of the public around, for example on farmland, access points to the area should be blocked and red-flagged.  The operator for the skeet thrower is situated so they are out of the line of fire too.  On pheasant shoots where the birds are of course unpredictable, each person is assigned a "peg" and told where they are allowed to fire, precisely to prevent falling-shot injuries.
It's worth noting though that all this still relies on having good discipline with a loaded weapon.  Accidents do still happen when skeet shooting.
While the lower terminal velocity of shotgun pellets may make them "less dangerous" than other projectiles, this does not make them safe by any means.  Using dimensions from Wikipedia and a terminal velocity calculator, and assuming a Cd of 0.1 for pellets (the standard figure for a smooth round ball), the terminal velocity of a BB pellet is 162mph, which is about the same as the fastest tennis serve ever.  Even for #10 shot, terminal velocity is 103mph, which is around the world record speed for a baseball pitch.  Also for reference, the muzzle velocity of a BB gun is generally above 60m/s, or 134mph.  Based on these numbers, the risk to life from falling birdshot is roughly equivalent to placing a BB gun against your head and pulling the trigger.
For completeness, I will note that @user560822 says they have obtained terminal velocity figures from the Chairgun app which are roughly half my values.  I don't have access to this app, or details of how the app does its calculations.  Airgun pellets are not smooth round balls, and I've seen references to bullets having Cd of 0.3, so maybe this would affect the figures from Chairgun and make them less applicable to figures for a shotgun pellet; or of course maybe my figures are based on an incorrect assumption.  I'll allow that my numbers above may be an upper value, but I can't easily find better data to make this more accurate.
A: Of course, responsible gun ownership is also to never fire up in the sky.

With that said, let's look at the physics aspects.
When a bullet isn't fired straight up the trajectory is in the shape of an arch, and generally the total time of flight will not be enough to lose most of the inital velocity to air friction. My understanding is that a bullet that is fired upwards at a angle of 45 degrees is still lethal when it returns to surface level.
It is theoretically possible, but in practice not feasible, to aim upwards so precicely that the trajectory remains very close to vertical all the way up. If it does remain vertical then the pull of gravity will remove all velocity. On its way down the bullet will regain some velocity, but that velocity hits a maximum pretty quick. 
If you would drop a bullet from a helicopter several kilometers up then for most of the way down that bullet would have the same relatively slow velocity. A bullet is not heavy enough: air friction makes the velocity top out at a relatively slow pace. If that bullet hits you on the head it is very annoying, but the hit won't be harder than being hit by a pebble falling from a couple of stories high.
The smaller the projectile, the more significant the air friction. 
The maximum velocity of a object that is dropped from height is called 'terminal velocity'. That is a bit of a confusing expression, the idea is that the 'terminal velocity' is the velocity that the object ends up with.
The heavier the object, the larger the terminal velocity. Conversely, the lighter the object the slower the terminal velocity.

The biggest risk, it seems to me, is failure to fire straight up, which is likely to happen. When the trajectory is an arch the time of flight is likely to be too short for the bullet to lose its velocity to air friction. 
A: I'm putting in an answer because, while the other answers are correct in the stated facts, they simply do not answer or clarify the OP's question.
The actual answer to your question OP, is simply:

*

*Yes, "falling" shot (or rifle bullets) are dangerous, indeed deadly dangerous.


*Quite simply, skeet shoots do have a safety area behind. The OP is completely, 100%, correct. You need a safety area where humans are not allowed.


*For shotguns, that safety area is about the size of two football fields. It's impossible for the shot to go further than that.


*A point of confusion. If you're not familiar with guns. In fact, rifles shoot incredibly further than shotguns.  A shotgun range is a couple hundred yards long. But a RIFLE RANGE is enormous - miles. (Often they back directly on to the ocean, or a desert.)
"Why is it apparently not dangerous to fire a shotgun (such as when “skeet shooting”) into the air?"
It IS very dangerous.  It is so dangerous, that skeet ranges are about the size of two football fields.  (That's the max distance the shot can travel.)
It's that simple!
A: I feel that while many of the answers here are good, they are missing a little something practical.
At a range I have visited I believe #6 shot diameter .11 inches is the largest allowed trap shot size, while #9 .08 inch is more of a standard. Out of a 12 gauge we are looking at 1 0z of shot at about 1000 fps
In comparison, let's take what I would consider the classic American rifle cartridge the 30-06 Springfield, created in 1906 this cartridge saw the US through both World wars and Korea; it's a bit of a beast.
Looking at interwar era M1 Ball (modern ammunition can be more powerful)
Diameter is .308 inches, and muzzle velocity is 2650 m/s the single bullet is .4oz
But the main issue is aero dynamics.
Look at the middle bullet there, that shape is designed to retain energy
https://en.wikipedia.org/wiki/.30-06_Springfield#United_States
On the other hand birdshot is a load of lead bbs.
https://en.wikipedia.org/wiki/Shotgun_shell#Birdshot
In summary the shotgun spreads out less energy among many relatively high drag projectiles whereas the rifle is putting quite a bit more energy
$$\frac{(0.4\,\mathrm{oz})\times(2500\,\mathrm{fps})^2}{(1\,\mathrm{oz})\times(1000\,\mathrm{fps})^2} = 2.5$$
into a single more aerodynamically designed projectile. The rifle retains it's energy much better. In fact people regularly have 1000 yard competitions with WW2 rifles and ammunition.
A: Well let me give you real live feedback.
I’ve been hit by shotgun pellets from skeet shooters down range. My local gun range had to change the direction of the shutgun shooters due to a newer development near by, and now some pellets occasionally land in one of the secluded area we sometimes use for setting USPSA practical shooting scenarios.
In that area it’s no uncommon to hear the pellets constantly raining and smashing everywhere around me. Chances of getting hit are still very rare.
We practical shooters that use that area, use baseball caps, SAfety glasses, long sleeve and thats pretty much, enough protection. I was told they bounce right away if they hit you in the hat.
I once got a direct hit, on soft tissue, around my fore arm. That one did sting like getting hit by a white plastic airsoft BB gun. After experiencing a direct hit, instead of a grace, I can testify that those tiny shotgun pellets have no way of penetrating human skin at very long distances, but it’s still not pleasant at all, and left a (very tiny) mark on the skin, which later healed.
Not sure how far away the shotgun area is, since it’s out of sight, but I would guess it’s  around 100-150 meters.
Shotgun skeet shooting pellets, shall by no means be compared to other ammo effectiveness at long distance. I would bet 223 should be letal if fired at 45 degrees anywhere it lands more than a mile away. Most ammo shot diagonally, beside birdshot pellets, would definitely brake the skin. I’ve once heard a engineer tell a story where a relative was struck out of nowhere by a 22 in the country side, he didn’t even heard the shot.(Story from votex optics employer)
My background:
I’m currently a PhD student in Civil/Mechanical Engineering, and practice practical shooting with people way more experienced than me. I’m no expert in ballistics, just general physics and engineering. English is not my first language.
