As part of a problem I've been set, I'm required to show that if $G$ is a compact group then there is a continuous group homomorphism $G \to O(n)$ if and only if $G$ has an $n$-dimensional (continuous) representation over $\mathbb{R}$.
One direction is easy: if $\rho : G \to O(n)$ is a continuous group homomorphism then it is a representation since $O(n) \le GL_n(\mathbb{R})$.
The converse is less easy (I think), and I can't really see where to begin. I'd like to find some continuous group homomorphism $\theta : GL_n(\mathbb{R}) \to O(n)$, but I'm not having much luck.
Any insight would be much appreciated.