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](//en.wikipedia.org/wiki/Hadamard_matrix#Hadamard_conjecture)); the above construction ($\cong$ [Sylvester’s](//en.wikipedia.org/wiki/Hadamard_matrix#Sylvester's_construction)) 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](//math.stackexchange.com/questions/3510189/give-a-general-class-to-which-a-specific-4-times-4-special-orthogonal-matrix).)