First - Do you have a factory service manual?
I would research the wiring diagram and test the power wires coming in and out of the ignition switch when the issue occurs.
Second - 70% of the time its the worn out ignition switch. You will have to use a dremel to carve slots in the screws holding it on to get them out. They use screws that the head snaps off from the factory for theft prevention.
Third - You may be experiencing multiple issues like the ignition module going bad or the ecu. Sometime Ive seen either overheat and cause this issue and once the car shuts off, they cool off and the car starts and runs fine for a while.
Its a start, let us know what you found.