Do we know what mathematician first considered, and perhaps named, what we call the group $\mathrm O(n)$, or $\mathrm{SO}(n)$, for some $n>3$?

I mean it specifically as group (not Lie algebra) acting on Euclidean $n$-space. For $n=3$ Jordan (1868) seems a definite upper bound, but for higher $n$ it seems not clear to me that even Cartan (1894) thought in those terms, describing as he does $\mathsf B_l$ and $\mathsf D_l$ as “projective groups of a nondegenerate surface of second order in spaces of $2l$ and $2l-1$ dimensions.” Also please disregard any implicit occurrence of $\mathrm{SO}(4)$ in quaternion theory.