You would not pick up a degree in the US (outside of rare double degree programs), but studying on exchange would let you indeed study in the US for a time.
I suppose the question is, outside of “studying in the US” (which you could accomplish on exchange, though look in to costs), what goals do you have?
BTW, the UC’s tend to have one central exchange program, which means you may not get assigned to UCLA even if you attend a uni with an exchange program with the UC’s.
If you really want to live in LA (for a little while), look to see what unis have exchange programs with USC (and other LA colleges) and how hard it would be to go on exchange.