You might be able to, but it would be up to the airline.
@KayaWildfire is right - too much potential for damage.
Suggestion 1. If you have your flight information already, visit
www.seatguru.com. See if your plans has a closet. Call the airline now and let them know you were bringing a wheelchair and would like to keep it in the cabin. They should let you know whether you can reserve that, or if you have to arrive especially early.
Suggestion 2.. Take the wheelchair to the gate with you, get a gate check ticket, take the wheelchair down the jetway, leave it outside the plane. The airline will stow it in the cargo hold, and bring it back up to the jetway when you land.