In my experience, spider bites look more like one bump with two barely visible holes in it. Like a mosquito bite, but two holes instead of one.
Also, I don't know what kind of spider you caught and killed, but my family had an infestation of some kind of biting spiders (they were black so probably not the same), and, trust me, there was no mistaking the bites. I won't go into detail, but my mom got one on her face, went to the doctor and he went "WOW that's ugly!"
But they needed no treatment (except for our poor cat who got bit, but when you only weigh five pounds, anything that bites you packs a big punch!), ultimately they just had to heal on their own.
Keep an eye on the suspected bite and see what happens, but even if you did get bitten, it's probably just your average 'round the house spider. If you start feeling horrible pain - the kind of pain where you KNOW something is up and don't have to wonder if you need a doctor - or feeling really sick (again, the kind of sick where you don't question if you should see a doctor) out of nowhere, it's worth getting checked out.
I should add that a friend of mine has been bitten by a brown recluse spider TWICE and lived to tell about it!