Preflop: I have AA on a $25 NL table

Pot is raised to .50, I call.

Looking at it after the fact I'm not sure re-raising would have made any difference in the outcome.

Flop: (board: 6c As 2s)

Bet of $1.35 is made to me, I raise to $5.40.

I raised here to push out a possible flush draw, wasn't really expecting a call.

Turn (board: 6c As 2s 4s):

We both check.

River (board: 6c As 2s 4s Qs):

He bets $4, and I fold after a long while.

Most of the time in hands like this I've found that the spade is out there, so I've learned to simply cut my losses. Is there anything I could have done differently here to figure out what he has? I could have bet on the turn, but a pot sized bet would be a big bet.

What would you all have done here?

He opted to show, I'll post later on what he had.