$SO(n,\mathbb{Q})$ 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.