Can I split off a pipe to create an additional faucet in my basement?
Turn off the main supply and connect the supply line(s) to the faucet. You will need a brass tee, crimps, and the connector for the pipe to the faucet, and some pipe. 

If you want to use a silcock instead of a sink, you still need to attach it to something. I would start there (build something to hold the silcock), and attach the pipe to it once I figured out exactly where I wanted it.