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:
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.