Do women approach and try to pick up attractive men the same way men go out and try to pick up women?

Be more interesting. Do interesting things, have interesting hobbies, read interesting books. Practice talking to people and girls in general. Talk about things that amuse or interest you, and do it twith energy.
That last part is extremely important. If you talk and do things with high energy, even if you're talking about dumb bullshit people get wrapped up in it. That's probably the most important part of picking up a girl. high energy=interesting and fun, signals confidence and authority.

This is pretty much it. I can always tell who is into me at the gym. They'll go out of their way to get in my way like when I'm walking to the drinking fountain or loading weights. And when we pass, they'll usually smile or awkwardly try to move around me while laughing, like they didn't just purposely plant themselves there to get my attention.

Most women try to maintain some kind of dignity so they very rarely put themselves out there and approach. Unless your Chad or Slayer mode

Robin?

>standing in your vicinity and ocasionally looking your way
yeah

>ask you some random question to get your attention
you mean some dumb, cliche thoughtless question that provokes the attention of her barely-sentient friends but scarcely elicits a response from you.

>hope you carry the conversation
you mean "expect you to carry the conversation". Great when their revolutionary best-shot icebreaker is a fucking yes/no question, or even a mere statement. "Do you work out?" "Yeah". "You have nice hair" "Thanks".

Honestly, I am one entitled bitch away from just laying down "k thx bby wan sum fuk?". Pretty damn sure they'll go for it, also pretty sure I'll hate myself and hate them even more for it.