Is a random Gaussian distributed vector rotational invariant?

Given $v$ a random Gaussian distributed vector, is it possible to prove that $r = Rv$ is also Gaussian distributed? Any formula-link-information on how to start working on it is highly appreciated.

Thanks.

Hence, for each component of vector $r$ we can write $$r_i = (Rv)_i = \sum_{j} R_{ij} v_j,$$ which is indeed Gaussian, given that $v_j$ is Gaussian.