How can I use bulk electrical wire to make a subfloor heating system?
Just don't. This is clearly the best answer.

If you really want to cheap out perhaps you could buy a complete kit with just one square foot mat, analyze the wire used, and try to source a designated radiant wire of similar dimension and resistance.  Then fold and spindle that in a similar way to the mat squares.

Remember your fire insurance likely excludes work done in flagrant violation of the building codes.  Use of regular copper wire, while possible, is just not advised.