Was it made with egg whites or meringue powder?
If it was made with meringue powder, you can store it in a Ziploc bag (with the excess air pushed out of it) or in a very clean Tupperware container (press plastic wrap right against the surface of the icing inside to prevent it from crusting.) These can be stored for long term in the fridge/freezer (though it's not my favourite solution for brightly-coloured/dark icings, I find they don't regain their consistency perfectly IMHO), BUT they can also stay out at cool room temp (in shade, not stored in direct sunlight/heat) for 4-6 weeks, no problem. There's no worry about mold, it contains nothing in it that will rot/mold.
You'll see that it will separate into three layers of liquid/solids/foam after a while....but it just needs re-whipping to use it again. If you think you haven't stored it well or for too long, just give it a sniff before you use it. If it smells slightly alcoholic, just throw it out. It's not so costly that you should ever risk using it if you think it's gone off. Just throw it out!
If it was made with egg whites, it needs to go into your fridge or freezer. Same thing though, it will need to be re-whipped before using it again.
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