Should I hang drywall up to the ceiling (using steel studs)?
Drywall is part of the fire retardation system in many types of construction. The thickness is specifically designated to provide a certain timing before a fire can spread to an adjoining area.

Failing to have the drywall fully enclose the wall defeats this. It also compromises the insulation of the room, both with regard to temperature and sound. It may also increase the likelihood of air infiltration (drafts and temperature issues).

If this is just temporary (as @TheEvilGreebo queries), you may be fine so long as an inspection is not done and you will seal up the space soon. 

If this is a permanent approach, you have a problem.