Best Storage:
Fridge
Ginger will last longer kept whole and unpeeled. Try to peel sections as you use them. In the fridge, keep it wrapped tightly in a beeswax wrapper or sealed container and if you cut sections, blot them dry with a paper towel before returning to the fridge.
Freezer?
Yes
Freeze whole and unpeeled. Place in an airtight container to prevent freezer burn.
Use it up!
Frozen ginger is easy to peel, so peel and grate what you need then return to the freezer.
Keep your peelings. Dehydrate them in a bowl on your bench and add the dried peels to soups or use to make ginger tea.