To the girl who wrote the above: did you show in any remotely visible way to the guy that you wanted to be picked up? Did you take out your headphones? Stop texting for a few moments? Look him in the eye? I doubt it.
Don’t beat yourself up too much. If the guy talked to you for a while, while sitting next to you on the plane and he didn’t ask you for your number, either he is not that interested, or didn’t have the balls for some mysterious reason to get your contact information, or both. In either event, you are most likely not missing much.