I think I'd like to be a project manager, but honestly I'd be happy doing it at my day job. Any suggestions for how to show people that you can do that kind of job?
I suggest checking out the book "The Passionate Programmer".
There's a general focus in that book about building your personal brand in an organization so you basically have leverage to do what you want.
Beyond that, the short answer is to assume the role. Take on a PM-like presence amongst your team if you haven't already - ie. help any PMs you're currently working with on small tasks, ease their burden where you can - then have a candid discussion with your supervisor at some point about how you can see yourself in such a role.
I totally agree with everything brandall10's written. I've always found that "act as if" can be a powerful tool, except perhaps in dysfunctional organizations. In other words, "act as if" you're already a project manager. And you'll definitely need to let your manager know about this career aspiration too.
If you want to do this as a remote contractor, then I think your options are:
1) Look for an existing dev agency that needs remote project managers.
There's a general focus in that book about building your personal brand in an organization so you basically have leverage to do what you want.
Beyond that, the short answer is to assume the role. Take on a PM-like presence amongst your team if you haven't already - ie. help any PMs you're currently working with on small tasks, ease their burden where you can - then have a candid discussion with your supervisor at some point about how you can see yourself in such a role.