Help with Quaternion.AngleAxis

So I'm trying to get my Camera to rotate around a single game object when I hold down the Middle Mouse Button.

My script works fine when rotating the camera around Vector3.up, but when rotating around Vector3.forward the camera moves up and down even though I am only moving the mouse down.

The problem I can see clearly, as the camera rotates, its "forward" rotation axis changes (this doesn't happen when rotating around "up" axis).

The solution? No idea, thanks in advance for your help! :)

Here's the code:

 private void Update()
     {
         if(Input.GetMouseButton(2))
         {
             camTurnAngle = Quaternion.AngleAxis(Input.GetAxis("Mouse X") * rotateSpeed, Vector3.up);
             camTurnAngle = Quaternion.AngleAxis(Input.GetAxis("Mouse Y") * rotateSpeed, Vector3.forward);
             camera_offset = camTurnAngle * camera_offset;
         }
         transform.position = targetShip.transform.position - camera_offset;
         transform.LookAt(targetShip.transform);
     }