You have a couple options oop.
1. Bet to take down the dead money when he doesn't have the ace. This turns your hand into a bluff though and you should give up if called. This is a good option, especially against nits and other weak/tight players.
2. You can 'check/fold and not tell anybody.' Since you're oop and a long way from showdown with second pair to the board, you're never winning a big pot here but there are reverse implied odds. This is good against players who float and barrel alot.
3. You can check/call and see what he does on the turn. This is a good option vs bad tags who c-bet then give up too much.

IP, I'd almost always check it back, keep the pot small, and try to get to showdown, playing the hand as a bluff-catcher. This works much better ip than oop because you have alot more control over the pot size. And it'll be more profitable than c-betting and giving up because people routinely stab after you check.