Laning against Aatrox
Aatrox is extremely reliant on basic attacks. Getting a Ninja Tabi and a Wardens Mail will shut him down HARD. Wardens Mail eventually can become a Frozen Heart or Randuins Omen, both of which are excellent defense items against Aatrox.
If his jump is becoming a problem consider a Phage to perhaps enhance your movespeed enough to catch up to him, or perhaps a Bilgewater Cutlass\Blade of the Ruined King. Both item's have an active that slows him down and the BoRK actually speeds you up as well.
A good Aatrox will make good use of his escapes, and short of using the assistance of a good gank, he may be hard to kill. The trick is catching him while his abilites are on CD as they tend to be moderatly long.
If you can't bring him down just farm it out, as long as you can get him to the point where he has trouble striking the minions, you've got him cold, as he can no longer sustain.
If you're ever having trouble top lane, don't be afraid to ask your jungler for assistance. If you don't make the jungler aware of a problem they likely won't see it and help you out.
As for the matchup, TL;DR Riven beats Aatrox in lane because he simply cannot trade with her. The key is understanding how to trade.
Riven straight up does more damage in a shorter time, and is free to engage/disengage with her high mobility, so Aatrox cannot win an all-out melee battle with her. Aatrox relies on auto attacks for his damage, and he should not be auto-attacking you more than once in a trade because you can Q-auto-W-Q-auto-Q-auto, which both stuns and slams him while dealing high damage.
His best chance at getting autos on you is committing with his Q so that you are cc'ed and he can start dealing damage. The problem with this is that, as stated above, Riven wins melee battles. He has committed to a fight he cannot win, and he cannot escape because his only escape is on CD and you have two short-range dashes (Q and E) to chase him with. His E is useless here because the slow does not reduce the chasing power provided by your Q and E. You will catch up with him and deal much more damage.
Since you destroy him in melee battles and he cannot risk a melee engage with his Q, you take the role as aggressor here. If he Qs away when you try to engage, you are winning. His Q costs HP, and the only way for him to heal is auto attacking. Keep doing this and he's just wearing himself down. Stand in the minions and zone him so he can't heal. Freeze your lane so no minions move closer to his tower. Punish him with your burst for coming too close - getting his Q on cooldown is sufficient. He cannot trade with you, cannot heal, and cannot farm, so this lane is completely over. You do not need to kill him, just establish dominance and farm because you scale well and can win other engagements with a gimped Aatrox.
There's only two things to worry about at this point: jungle ganks and Aatrox' harass. As for jungle ganks, ward up the river. You can easily back off when the jungler comes, or even kill Aatrox if his passive is down. As for harras, Aatrox only has E, which costs him HP and is extremely easy to dodge. Keep in mind that you are Riven - you have high enough base speed to move out of his E skillshot, and you also have two short-range dashes (Q and E) to help you out if you are having trouble.
You really don't need to build a whole lot against him - Ninja Tabi are sufficient or get another armor item. Build as normal and you'll continue crushing him in lane.
If you want to get a feel for how this lane works, watch WingsOfDeath play Riven against Aatrox. Unfortunately I lost the old video was taken down, but this one still demonstrates the lane.