How can I make a camera with a full 360 degree view on all axes?

I'm currently helping out on a Roblox game called Starscape [edit: Evolution Online. Accidentally added the name of a different game]. I'm in charge of the spaceship functionality, and have been recently making a lot of progress, with a lot of movement, thrust, and rolling functions working correctly. However, one thing I've not yet been able to get right is the camera.

While I've scripted and tested many different camera systems at this point, many of which have allowed smooth and easily controllable spaceship movement, all of them have had some disadvantage meaning that they could not be used.

For example, one of them could rotate freely on all axes, but the mouse movements were inverted when upside down; multiple others had the correct mouse movements at all angles but only allowed looking up or down in terms of one of the axes (wasn't able to go upside down); several others just never worked correctly.

I would really appreciate if someone could help me out with this. I've spent days working on this, and have had to constantly go back to square one - I've only just finished yet another system, only to discover that it had to be discarded due to be axial limitations. I just need a general camera system or at least some guidance; I can sort the rest out myself.

Thanks in advance, Dominic