Yes, if you use all-purpose bathroom cleaner and a lot of elbow grease, you can rid your shower doors of soap scum, eventually.
Or, you can use vinegar. It cuts through that film like a knife through warm talc combined with body oils in a steamy environment.
To remove the chalky streaks, dip a sponge in full-strength, white vinegar. Coat the door, and keep applying as the vinegar dries. You want to keep the door covered in wet vinegar for at least 5 minutes. The film will melt away. If you have particularly stubborn soap scum, you can try applying boiling vinegar -- very, very carefully.
Rinse off the vinegar with warm water, and you're done. The vinegar smell will go away after a couple of showers.
Other options include orange-based cleaners, mineral-oil solutions and liquid-fabric-softener solutions.
Once the scum is gone, switch to liquid soap, which is lacking the talc that can lead to soap scum in the first place.
Up next: A shower-door health hazard, and how to eliminate it.