If $G=SO(n)=SO(\mathbb R^n)$ and $r\leq n$, it is easy to find a closed subgroup $H\leq G$ that is isomorphic to $SO(r)$, just let $S\subseteq\mathbb R^n$ be an $r$-dimensional subspace and let $H=\{g\in G:(\forall x\in S^\bot)g(x)=x\}$. These are the trivial examples.

If $r=2$ and $m\geq 4$ we can find nontrivial embeddings $SO(r)\to SO(n)$, for example:

$\begin{pmatrix} \cos t&-\sin t\\ \sin t&\cos t\end{pmatrix} \mapsto \begin{pmatrix} \cos(2t)&-\sin (2t)&0&0\\ \sin(2t)&\cos(2t)&0&0\\ 0&0&\cos(3t)&-\sin(3t)\\ 0&0&\sin(3t)&\cos(3t)\end{pmatrix}$

Are there examples for $r=3$ (or more)?

non-closedsubgroup of $SO(n)$ be isomorphic to $SO(r)$?) – Mariano Suárez-Alvarez♦ Jan 25 '13 at 19:54areseveral classes of embeddings, even up to conjugatio. The classification is, as Claudio notes, the classification of faithful orthogonal reps of $SO(r)$ of a given dimension. – Mariano Suárez-Alvarez♦ Jan 25 '13 at 19:57