Organizational skills are key to becoming a project manager, but they're sufficient. The most important thing to have is "people skills".
Apparently, people are happy to work with you, which is good, but you need to:
- have the ability to make them respect you and follow your lead even without authority (of which you usually have none).
- respect people yourself.
- be able to resolve conflicts between people (being completely objective and not biased to any person is a good start).
- be able to say no to your superiors (you don't want to be tricked into committing to an unrealistic schedule).
You must also possess some "personal skills":
- Able to admit your own mistakes.
- Non-judgmental (people hate dealing with judgmentals)
- Constant work on yourself to enhance your project management skills, including attending courses, seminars, etc...
You have also understand and accept that project management will suck out your life; the line separating your personal life and work will start to fade.
From what you've mentioned I do think that you can become a project manager, just read the above and see if you meet the criteria mentioned.
One last thing to mention, if you find it hard to get promoted to a project manager position in your company, try looking for a junior project manager position in another company.