Can I attach 2x4s to the studs in order to mount my TV?
This is actually one of the most often recommended approaches to solving this problem. If you're up for a bit of drywall work. You could cut open the wall a bit, and install the 2x4s between the studs. 

For a quick fix, bolting 2x4s to the studs on the outside of the wall should work just fine. If you paint the 2x4s the same color as the wall, it might help to blend them in a bit.

Another option is to try and find a mount that accommodates 24" stud spacing, though this may not be an option if you already have a mount you want to use.