How to rotate a object without changing the axis by SIO2?

I am now doing a project requiring to rotate a cube freely in X, Y, and Z direction,
but I found that while X and Y axis will chenge when retating the cube, the Z axis will always points to the same direction,
whitch in resault causing a lot of problems.

Is there any posibility that I can fix all the three axises( x, y, z) while rotating by a cube?

A bit of linear algebra will do the trick

Sorry, I still can't really get the picture.
I look throw some .h files of SIO2, and I noticed that the rotation in SIO2 is implemented by "Euler angles" ( please let me know if it is not correct), which can cause "Gimbal Lock", and "Gimbal Lock" is also a problem that I faced.
So is there a solution really requiring "a little linear algebra" that can fix my problem?

Maybe sio is referring to quaternions?

http://www.flipcode.com/documents/matrfaq.html#Q35