OK, I already know the short answer, "go see a doctor"

A few months ago a noticed a couple of little spots on my right fore arm and another one on my upper arm. They looked like pimples or mosquito bites and itched slightly. The itching stopped after a couple of days, BUT the spots are still there! They don't itch or hurt and they have almost healed on their own, but they won't go away entirely.

Does anyone know that they might be? Has anyone had anything similar?

If that is you in the lip lock (you look old enough to be aware of pre skin cancer possibilities) would have them examined and probably removed by a good dermatologist. I have recently used Dr. Rachada at Vejthani Hospital for removal of growths on both arms and my neck and highly recommend. Simple, painless and not expensive (for first class equipment/facilities) and a doctor that takes the time to do it right. Treatment was far better than the most known hospital where I had similar treatment a year earlier.