How to run a cable underfloor from a wall socket?

Yes, as long as you do not run it under the floor joists. New electrical code requires that you drill a hole through the floor joists and run the wire through those holes. It cannot be hanging under the floor joists. If you see wires are already running under the floor joists then go ahead and run the wire under the joists attaching it on every other joists with a wire staple. Make sure they wire does not make contact with any metal ductwork or any other metal objects. Match the wire with what is already in the wall outlet you are jumping off of. If it is 12/2 then use 12/2, if it is 14/2 then use 14/2. Never mix wire gauges.