I think Velcro would be better than elastic. Elastic will sag no matter how tight you pull it. I just did this(with elastic)on a rolling table my DH made me. I wanted to use a curtain spring, but I can't find them anymore. I guess they are no longer being made...it's just a long spring with loops on the ends. Anyway, I had to put a center(hook)support up for my elastic and it doesn't look very good, so I'm thinking of putting Velcro along the edge instead...just haven't gotten around to it yet.
If you use Velcro, you'll have to have a stationary gather(and maybe two separate pieces for a split in the middle), so it will be a little more time consuming, but I think it will look softer than a straight/flat skirt.
What kind of seat do you have?

Thanks, yes I will try velcro. The seat is a vanity stool with a round cushioned seat and wrought iron legs. Not sure if I will make the skirt to just cover the seat or to the floor.

You can buy velcro that one side is sticky back and the other side is sewable (no sticky back).
Kathy G in MI

I have used the sticky velcro to put a "skirt" around a filing cabinet that need to be in my bedroom. I then just put a table topper on top and used it like a table even though it was a tall cabinet.
Hope this helps.
Sally

I will give it a try and let you know, thanks so much.