It's a great idea to send them a new version of your resume to start a dialogue again and to highlight your new skills. If it were me, I would reach out to the person I talked with before and let him/her know that I've remained interested in the company and just wanted to check in. If the person is a recruiter, they will probably immediately start thinking through whether or not there is an opportunity for you there.

I've had two experiences with revisiting a company after having turned down a job offer. A few years ago I had two offers at once. I took company A's offer and said no to company B. A year later, company B reopened the job I had previously turned down, and I contacted them. I re-interviewed and they made me a new offer which I accepted, and I was there for almost three years. Very recently, I turned down a job offer from company C. Although it felt right to turn down the offer, a week later I had doubts about it. I read up on similar situations from the recruiter/hiring manager perspective and many of them seemed open to the idea of having their top candidate change their mind. So I called the recruiter and said that after much thought, I would love to work for them if they would have me. I let her know that I understood it was a long shot, but I had to ask. She was very nice about it, but they were no longer interested! In fact, the job is still open.

It can go either way, but the old saying "nothing ventured, nothing gained" definitely applies. Company B welcomed me, while company C doesn't want to bother with me. It comes down to the perspectives and personalities of the people involved. In both cases, I'm really glad I re-approached both companies.