Hold your hands up out in front of you palms out so there is a small triangle or hole in the middle. Look at something far away and slowly pull your hands towards your face. Which ever eye your hands go to is your dominant eye.
Another way is to hold your hand out in front of you at arms length with your index finger up. Look at something with your finger in the way. You should be able to "see through" your finger with both eyes open. Cover one eye and see what happens, then the other (seperately, duh, don't cover both eyes at the same time). One eye will make your finger cover the object you are looking at. The other eye will make your finger shift to one side. The eye that makes your finger cover the object is your dominant one.
I suggest shooting with your dominant eye. For instance I am left handed but right eye dominant. I shoot guns, bows and throw right handed. I can actually shoot a handgun with either hand. Most left handed people are ambidextrious. Sort of a perk for being screwed on everything else

.
Paul