How thick is the "skin" formed from surface tension? I learnt in class that surface tension is caused by an unbalanced force at the surface of the liquid due to IMFs, forming a "skin" on the top. Does this mean that the skin is just one molecule thick? My teacher conjectured that there might some sort of gradient, with stronger forces at the top gradually decreasing. Is this the case? How thick is it? Please correct any misconceptions that I have.
 A: This review may give you a feel for the factors involved.  They are looking at the ion concentrations at the water liquid/gas interface, and they cite a number of studies tackling the problem from various angles.  You'll see that the characteristics of the water are changing over a length scale of 10-20 angstroms.  Water molecules are more like 2-3 angstroms, so that is several molecular layers thick.
Another study using Small Angle X-ray Scattering looked at an (exotic) organic liquid and found ordering up to about 10 nm deep.  In this case, the molecules were polar, and prone to ordering so the surface didn't evidence a a gradual drop off of order, but rather there were about 4 molecular layers that showed ordering, where each layer had well defined units, and below that random liquid.
Thus, the exact answer to your question depends on the specific liquid, pressure, etc.  However, the magnitude of this depth is in general "a few molecular layers thick" and may or may not be a gradual gradient of any one property (ordering, density, solute concentration, etc).
A: As a first-order approximation you can think of surface tension as affecting only the top layer of molecules. I'm sure that a careful molecular dynamics simulation could show some subtleties, but the basic length scale involved is one layer. There's an interesting justification for this in 
"Search for Simplicity", Victor Weisskopf, Am. J. Phys. 53, 19 (1985)
http://www.pha.jhu.edu/courses/171_602/weisskopf_simplicity_long.pdf
To summarize, suppose you measure how much heat it takes to vaporize a certain quantity of, say, liquid methane. Assuming you know the molecular weight, you can turn that into the heat per molecule for vaporization. Next, assume that vaporization breaks on average three bonds per molecule. (Each molecule participates in six bonds: above, below, left, right, front back. Each bond involves two molecules, so that's three bonds per molecule.) This gets you the energy per bond.
Next, go back to the liquid and look at molecules on the surface. Assume that they participate in five bonds, not six, since the one at the surface is missing. If you know the size of the molecules, you can calculate the number of surface molecules per unit area, and using the bond strength you can get the energy per surface area. This is the surface tension.
It turns out this simple model works well: to within maybe 10% or so unless something funny is going on (like the molecule being highly polar so that surface molecules can flip around to maximize bonding). The accuracy of this simple model justifies the idea that surface tension can be accounted for primarily in the single top layer of molecules.
