if memory serves, he drank himself to death, much like what Brick is doing. i believe there is a line uttered by Brick or Maggie in Act I saying "And poor Skipper drank himself to death."
he definitely did not jump off a building or commit an overt suicide, that is specifically referenced. he let himself waste away/tore himself down with excessive alcohol due to his unrequited love for Brick.
"I am not 'a' Eunice Burns. I am THE Eunice Burns!!!"