Dependence of friction force on contact area between skis and snow According to Amontons' laws of friction the force of friction is independent of the apparent area of contact.
But from my personal experience with skiing I know that skiing on the edges of the skis you gain more speed than skiing on the whole surface of the ski. And the difference is not marginal but substantial.
That is why professional skiers try to stay on their edges most of the time on a ski trail. And that is also why, not a long time ago, skis with straight edges were replaced with curved edges - because then you can stay on edges for a longer time even on turns.
I tested it on soft snow and on snow almost hard as ice - in all cases skiing on the edges was faster.
Because of that the fastest way from point A to point B on skis is not the straight one but a curved one. There is some optimal curved trail - of course if you do some extreme curves you will not be faster than on a straight trail.
So my question is why it is so? The only difference between the two skiing techniques I see is the difference between contact area. But the friction force should be independent on that area.
P.S.: Assume steel edges and the rest surface of skis have the same friction. It works even with a non-steel edges, so the different material of the edges is not the reason why it is so.
 A: 
So my question is why it is so? The only difference between the two
  skiing techniques I see is the difference between contact area. But
  the friction force should be independent on that area.

I'm  no skier, but could it be that skating on the edge of the skis makes them cut through the snow better at the leading edge? I mean, causing less "plowing" of the snow in front of the skis.
You may want to research the possibility of a different coefficient of kinetic friction for the snow that would be more "packed" under the narrower surface of the edge of the ski. But I'm not sure which way that goes (higher or lower). 
Hope this helps.
A: The smaller surface increases the pressure on the snow which melts it more so the think layer of water, which acts similar to a lubricant, is thicker and reduces the friction. This do not happen in friction of solid against solid, unless you purposeful add water as a lubricant.
A: This friction model is known as "dry friction".  To quote wikipedia's article on dry friction:

Dry friction resists relative lateral motion of two solid surfaces in contact. The two regimes of dry friction are 'static friction' ("stiction") between non-moving surfaces, and kinetic friction (sometimes called sliding friction or dynamic friction) between moving surfaces.

(emphasis mine)
This relates to ja72's comment and Bob D's answer.  The interaction between the ski and the snow is not really accurately modeled as "lateral motion of two solid surfaces".  It's much more complicated than that.
The edge or surface of the ski doesn't just slide over the surface of the snow; if that were the case, you might be able to expect dry friction to hold fairly well as an assumption.  Instead, it actually changes the surface of the snow as it goes; which leads to complex internal interactions of the snow as it piles up and moves around itself and the skis.
In this way, it behaves somewhat like a fluid.  The less it impacts the surrounding snow, the less resistance from the mass of snow that you need to displace as you cut through it.
It's also worth noting that it's not just about staying on the edges of your skis.  The edge of your skis also have to be parallel to your direction of travel.  This minimizes how much snow you "plow" with the skis.  The plowing gives very significant resistance to motion; which is why you typically dig your edge in perpendicular to the direction of travel when you want to stop.  This effect becomes quite apparent when you try and stop on an ice patch as compared to loose snow.
A: I think the reason is that when skiing, the friction is not only due to ski/snow contact but also about the fact that snow needs to be compressed in front of the ski. Having 2 skis requires you to compress and push more snow in front of you.
It’s not possible for the snow to melt under your ski because the pressure is by far not high enough.
