You can endorse a check over to another person.
Technically, the paying Bank can demand proof that the original payee of the check endorsed the check. Just about the only way to do this is by having the original payee physically appear at the teller line with an ID.
But depending on the Bank, the Bank may not require such proof. If you get away with it a few times, fine. But don't be surprised if the check gets returned unpaid someday. As cerebusPu pointed out, anyone can fake an endorsement, and the payor Bank will have to take a loss if it accepts a check with a forged endorsement.
It's better just to have the payor write a new check.