I don't like to poison animals unless there's no other practical way. It's too slow for me. I cant stand the thought of an animal dying in pain like that (unless there's no other way - except for mice and rats... for *me* there's no other way).
Anyway, back to your skunk. If it were me, I'd trap it and shoot it. Since you do not own a gun, I'd recommend asking someone who does own a gun to come and shoot it for you. Or call animal control. It wouldn't be beneath me to call the sheriff's office and ask if someone could come out and shoot it if they're not too busy - obviously they have more important things to do, but if they can, they probably will help you out.
If those routes don't work for you, I'd submerge it in a big huge garbage can full of water (which I consider too dangerous myself, but you can do it with rods and poles...in other words, don't be putting your hands on the trap to move it or you can get clawed at) or a running creek and just go away and come back later. Obviously, if your trap is small enough for you to lift by a handle, that would work. I'm just saying take care of your fingers. Besides, you might get sprayed by moving it unless it's covered by a tarp as Huntertima suggested.
Keep in mind that if you set out poison for it, you can poison other animals besides the animal you wish to poison. So take real care with poisons. If you absolutely must use poison, I'd use it only in the trap that way you know only your target animal is poisoned and it doesn't run off to be poisoned food to another animal.
Good luck catching your skunk! Be careful too!