I've liked this guy for a while now. I think he likes me, too, but I'm not really sure. We've been flirting a lot. My friends say I should just tell him, but I'm old fashioned and romantic. I think a guy should approach the girl. Is there a way of knowing if he does like me even though he isn't asking me out?
You're not old-fashioned – you're smart. If he doesn't ask you out it is for one of two reasons:
1) He's not interested.
2) He's shy, unsure, or lacking confidence. This does not mean he needs a little push from you or any other woman. This means he doesn't have it to push himself.
While a shy or unsure man may sound like a dandy fix-it project for some women, for others it is an alarm. He lacks confidence and isn't willing to take the least risk to get what he wants. Later (in marriage, with children, and with job prospects) this can translate into unhappy relationships, abandoned children, and un/under-employment.
Throw hints at him until you're done doing so. If he doesn't take the bait, move on. Don't waste too much time on him, lest you walk right past the guy who really is willing to go on a limb for you.
Do be aware you may be masking your own insecurities and lack of confidence with terms like "old fashioned." This isn't likely given the amount of flirting you've mentioned, but just in case, be mindful of your willingness (or lack thereof) to drop those hints and give him the go-ahead.
It's worth noting the amount of concern you have for what he's doing rather than what you're doing. Do remember, the most effective way to attract the man you're looking for is to stop looking for him and invest in your own interests. A church-goer won't stumble across the man of her dreams in a bar, and a theater buff won't find Mr. Right at the laundromat.
Nothing is more attractive than someone enjoying themselves – so get out there and do your thing.