Let $n\in\mathbb{N}$. Let $\rho:O(n)\to GL(\mathbb{R}^n)$ be the standard representation, defined by sending each orthogonal matrix $A\in O(n)$ to the linear automorphism $\rho(A):\mathbb{R}^n\to\mathbb{R}^n$ given by $\rho(A)(x)=Ax$. Then $\rho$ is injective.