$\DeclareMathOperator\SO{SO}\SO(n,\mathbb{Q})$ is the group of $n\times n$ matrices $A$ with rational entries such that $AA^t=I$.

The $n$ coordinate subgroups of $\SO(n,\mathbb{Q})$ are the subgroups $H_i$ consisting of those matrices that fix the vector $e_i$, where $e_1,\ldots,e_n$ is the standard basis for $\mathbb{Q}^n$. Each such subgroup is isomorphic to $\SO(n-1,\mathbb{Q})$.

For which values of $n$ is $\SO(n,\mathbb{Q})$ generated by its coordinate subgroups?

Stefan Witzel has shown me a proof that this doesn't happen for $n=3$ and some other results, but I will let him describe these.