There's a structural difference between the rotation groups of $\mathbb{Q}^n$ and $\mathbb{R}^n$; in some abstract sense the former is 'small' (discrete?) while the latter is 'large'. I suspect that part of the root of this is what might be called distance-closure: the distance between any two elements of $\mathbb{R}^n$ is an element of $\mathbb{R}$, whereas distances between two elements of $\mathbb{Q}^n$ aren't necessarily in $\mathbb{Q}$; in essence, rotations in $\mathbb{R}^n$ are transitive on $\mathbb{RP}^{n-1}$, whereas rotations in $\mathbb{Q}^n$ aren't transitive on $\mathbb{QP}^{n-1}$ (for instance, there's no rotation of $\mathbb{Q^2}$ mapping the line $\langle t,0\rangle$ onto the line $\langle t,t\rangle$).
But there are intermediate fields between $\mathbb{Q}$ and $\mathbb{R}$ that are still distance-closed; in particular, the (real) algebraic closure of the rationals $\mathbb{\bar{Q}}\cap\mathbb{R}$ (I'm going to call this guy $\mathbb{\hat{Q}}$ in lieu of any better symbol) is. Virtually all of classical geometry translates directly to $\mathbb{\hat{Q}}^n$, and we get the transitivity of rotations on $\mathbb{\hat{Q}P}^{n-1}$ mentioned above. Is anything known about this group of rotations, and in particular — while it's obviously not as topologically nice as $O(n)$ since $\mathbb{\hat{Q}}$ isn't complete — is there a good structural sense in which it's 'larger' than the rotation group of $\mathbb{Q}^n$, beyond just the transitivity I mentioned above?