Best way to store ginger is too chop it up into walnut sizes and freeze in a bag. When you want to use it, take a piece out the freezer, leave it until it is a bit defrosted and then grate it. You can leave the skin on.
I have always peeled it when using fresh ginger. However, as there is usually far too much for me to use for one recipe, I freeze it in one piece, then grate it from frozen as and when needed - and I don't peel it.