I have 3 black skirt tetras in a 10 gallon but I'm not positive that this is big enough for them? Do they need more space to swim and do they need more friends?

Yes, you're correct in thinking they need more space. They need a 2 ft long tank at a minimum, so at least a 15 gallon but preferably a 20 gallon long which would provide much more swimming space than a 15 gallon or a 20 high would.

And yes, black skirts are schooling fish so they should be kept in groups of 6+