Rotating other 3D vectors based on one vector's image

Question

I have an initial, normalised 3D vector $\vec n$ and its result after an unknown rotation, $\vec n'$. Given another normalised 3D vector $\vec \omega$, is there a vector formula for getting its image $\vec \omega'$ under that same rotation? I'm imagining that mapping $\vec n$ to $\vec n'$ is like rotating the unit sphere, which should allow easily remapping every vector on the sphere. On the other hand, since we're rotating, it probably isn't as easy as doing a projection, unlike e.g. the vector formulas for the law of reflection and refraction.