If you like dealing with clients (and it sounds like you may be patient enough to be good at doing this), you could try your hand at being a project manager and managing several development contractors for the projects you get from your clients. I built a web development agency like this several years ago.
Basically, you start with the same clients that have been hiring you, then tell them you can assemble a team of developers to help get their project done, and all they have to do is work through you as the main point of contact.
This is easier said than done, of course. Starting with existing clients is a nice way to jump-start this kind of a business, but you'll need to find more clients eventually. There's also a whole arena of skills you'll need to develop in terms of recruiting, account management, and project management too.
This isn't for everyone. But someone with technical skills has a vast advantage in such a role over someone who doesn't, with all other skills being equal.
Just an option to consider as you think about moving out of a technical role.
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.
Basically, you start with the same clients that have been hiring you, then tell them you can assemble a team of developers to help get their project done, and all they have to do is work through you as the main point of contact.
This is easier said than done, of course. Starting with existing clients is a nice way to jump-start this kind of a business, but you'll need to find more clients eventually. There's also a whole arena of skills you'll need to develop in terms of recruiting, account management, and project management too.
This isn't for everyone. But someone with technical skills has a vast advantage in such a role over someone who doesn't, with all other skills being equal.
Just an option to consider as you think about moving out of a technical role.