When you turn the ignition to on, when starting the vehicle, the AirBag module sends testing voltages through the AirBag circuits. If any of the circuits have a fault, or any part of the system is missing, then the AirBag light is illuminated and stays on as long as the ignition is on. Removing the steering wheel AirBag to fit the alternative wheel should illuminate the AirBag warning light continuously if the circuits are not good. A primary checking routine would be to disconnect both terminals of the battery at the same time. Let the vehicle stand for around ten minutes, or more, with the battery disconnected. Now check all of the cable plugs on the steering wheel AirBag circuit. Disconnect them, spray with electrical cleaner and allow to dry. Reconnect all plugs and reconnect the battery. Turn on the ignition and check for correct operation of the warning light. If the light remains on, then the choice you have is to have a code reading by either buying a code reader for AirBags for your vehicle, or a local parts store/repair shop do a reading for you. The prime suspects in your case are plugs not connecting properly, or the vehicles 'clock spring' has become damaged and open circuit. This can easily done if when fitting the steering wheel parts, the steering wheel and the clockspring assembly are not exactly in the straight ahead position. When testing any part of an AirBag system you must not use an Ohmmeter. The voltage of the Ohmmeter can and will set off AirBags or ignitors by their voltage. You can substitute any part of the system by placing resistors in the circuit to 'prove' a components condition and reconnecting the battery and turning on the ignition as usual. If you have to go this far though, you really should have a repair shop do the work. Here in the UK some shady used car dealers have been taken to court because they used resistors in the place of new conponents.