I'd wear a coat you can always. Check it there if you don't want it.
Its usually very cold down on the ice. So as mentioned before its best to dress warm with something you can remove if you get too hot.
That's up to you just remember that your skating on ice so it's cold in there. But if your skating and raising your heart rate lighter layers might work but I would definitely dress in layers thinking it's going to be cold in there.
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