To render the skybox, I use the same technique that OGRE uses in its sample Sample_TutorialSky_Postprocess. As the title says, I'm wondering if there's an easy way to apply a rotation (to make things simpler: just a yaw around the world y axis) to the skybox.
I know I can simply apply a counter-rotation to the objects, but doing this creates a mess with the AABB visualization.

I don't know if you need to swap the multiplication order though (i.e. try float3 * float3x3 if float3x3 * float3 doesn't work as intended or rotates in the opposite direction).

xrgo's answer works on rotating with 2 axes by using a float2x2, instead of 3 axis; but is basically the same concept.

I don't know if you need to swap the multiplication order though (i.e. try float3 * float3x3 if float3x3 * float3 doesn't work as intended or rotates in the opposite direction).

Yep, I had thought about using a rotation matrix in the vertex shader. Correct multiplication order is float3x3 * float3.