Why do Predators flay people? And what do they do with the skin? We know that they collect skulls, but why skin?
The Predators were supposed to be a metaphor for human sport hunters. In fact that was the working title according to Wikipedia:
Predator was scripted by Jim and John Thomas in 1985, under the working title of Hunter.
As we know, people who hunt for fun typically take part of their prey's body as a trophy, usually skulls, but also skin (i.e. leather) either stretched or stuffed, paws, tusks, etc. depending on the animal.
So, if human hunters keep animal skulls and skin as trophies, it isn't too inconceivable that humanoid aliens would also do so.
I propose the skin is not taken as a trophy, there it lies beneath the hanging human carcass.
Notice when the skinning takes place: at the beginning of the hunt.
In Predator and Predator 2, both have occurrences of the Predator on the hunt making a point of making a hanging display to its intended prey as they enter the hunting territory.
This probably is inter-species communication, as a warning to the humans that they are entering the hunting area. The Predator probably does this to divide the choicest trophies from the ones which would be turned away by such a gruesome sign post.
How I draw this conclusion is by how the Predator on the hunt, even further evidenced in the movie Predators, chooses its intended prey, it doesn't take trophies from pregnant women and the unarmed, it wants hunters to hunt, seasoned killers.
Those who gaze upon the skinned warrior and do not turn back are more than likely a blooded, hardened warrior or not fit to breed more potential prey for the Predator to hunt by virtue of being a fool who ignores the signs that a hunt has begun in the area. Even then, those not armed or children that find them self in the hunting territory are given a chance to leave.
Well, Predators who collect the skull as a trophy might actually be winning chances to mate (it's been proven once). But as for the skin, they might use that for a carpet or stuff the skin like we do with deer. In other words the Predator likes winning the favor of the only female in the tribe and likes the skin for a carpet or a stuffed mantle piece.