I'm pretty much ambidextrous, although I do have to learn to do things with each hand if I want to actually be any good. It takes a bit of getting used to that I'm using my right eye to do things with my left hand. I can write legibly with both hands, although if I start thinking about it, writing left-handed gets to be a real mess. But, as long as I'm not thinking about it too much, it's better than most people's dominant hand.
That being said, I do have the concept of handedness for most tasks. It's mostly because I have a definite dominant eye and the practice it takes to have hand eye coordination.