Simple enough concept, PacMan with guns. Okay, sign me up.
There's the base of a really fun game here. Unfortunately adding a gun without adjusting the mechanics really ruins the challenge of PacMan. Maybe have the skeletons be more of an immediate threat? After all, the ghosts in PacMan will kill you instantly. You should also consider tying the ability to use the gun into a "powerup" like the cherry in the original PacMan.
Game suffers from a lack of "ending" besides game over, and no real sense of achievement for picking up gems. It seems like the skeletons are gathering gems too, but I can't really discern a reason to STOP them, ya know? Just let them have their fun. I'll just chill in the corner not getting noticed. You do you, skelly-dudes.
The absolute most important critique I have is that you have to cHanGE tHAt FUCKinG SOuND.
BELEpp BLopP BleoiuP Blopapi BLAHHGP MAKE IT STOP, good lord! Even with pitch variation, that sound effect on gem pickup is grating enough to inspire minor headaches, no joke. I couldn't play more than 10 seconds without giving up on the main objective.
Complaints aside I was pretty impressed by your AI for the skeletons, as well as the clever spawning system which ramps up the difficulty. I think the doubling and respawning was a superb idea.