Heat loss from skylights
Being that this is an apartment, you probably would need permission from the landlord, but a relatively simple way to solve this problem is by adding a frame with a lip on the inner perimeter of the skylight shaft as ddawg16 suggested, and then add some decorative framing on the ceiling to finish it off. Get a plexiglass grate and cut it to fit so it sits on top of the inner frame lip, and then cut a piece of clear plexiglass to fit and lay this on top of the grate. I had the same problem that you do and that is exactly how I took care of it. Before I did this, just standing below the skylight during the winter and you could feel the cold air falling down on you from above, plus the condensation on the inner surface of the skylight itself would often drip onto the floor below. Now, there is no more cold air from above and the condensation problem is gone.