Let $J_x, J_y, J_z$ be the generators for rotation about the $x, y ,z$ axis. That is, $\exp[{\theta J_i}]$ is a rotation about the $i$ axis by an angle $\theta.$ Furthermore, $\exp[{ \theta_x J_x + \theta_y J_y + \theta_z J_z}]$ is a rotation about the vector $(\theta_x, \theta_y, \theta_z)$ by an angle $|(\theta_x, \theta_y, \theta_z)|.$
Does this relation hold more generally, or it is a consequence of the coincidence that in three dimensions a rotation in a plane can be identified with an axis? For example, for $SO(4),$ we have six generators $W_{wx}, W_{wy}, W_{wz}, W_{xy}, W_{xz}, W_{yz}$ which are each a rotation in the $ij$ plane. What would $$\exp{[ \theta_1 W_{wx} + ... \theta_6 W_{yz}]}$$ be? Is the relation in $3$ dimensions just a general fact about Lie Groups and Lie Algebras, or again, does it only hold in $3$ dimensions?