If I have an orientation defined by Euler angles and I want to simulate a rotation of the coordinate system about the origin (doesn't matter to me how the rotation is specified), how would I get the new Euler angles? I understand how to transform a point via a rotation matrix, but I'm not sure how to approach this in terms of Euler angles.
For example, I'm thinking of an airplane's orientation in terms of pitch, yaw, and roll, and say I want to simulate a random rotation of the airplane about its center.