If your roof material is fiberglass or aluminum, you can easily seal around the skylight so it never leaks again - unless the sun eventually cracks the skylight material itself such that it's not a joint sealing issue (however, 1' or 2" EternaBond can of course also be used to permanently seal cracks in the skylight material) . You don't need to mess with a chaulking gun at all.
Just lay down a 4" wide strip of EternaBond tape around the entire skylight's edge ... about 1 1/2'' onto the skylight edge material and about 2 1/2" onto the roof material. You don't even have to remove any existing chaulking. 4" wide EternaBond will cover it all and still keep all water out for a long, long time. I used white colored EternaBond because our fiberglass roof is white.