Well, if $Q\in\mathrm{SO}(n)$ satisfies your (original) conditions then so does $ \frac1{\sqrt2} \begin{pmatrix} -{}^tQ&Q\\ \phantom{-}{}^tQ&Q \end{pmatrix} \in\mathrm{SO}(2n). $
Added: As S. Stadnicki since commented, your desired set $\mathrm S$ of possible orders $n$ is contained in and conjecturally equal to $\{1,2\}\cup4\mathbf N$ (Hadamard conjecture); the above construction ($\cong$ Sylvester’s) just shows $\smash{2^{\mathbf N}\subset\mathrm S}$. (Voting to close, as R. Israel had really said all this at the mis-linked question.)