Suppose that $\mathbb R^n$ is a module of the Clifford algebra $C_n$, with generators $e_1, \dots, e_n$, and relations $e_i^2 = -1$, and $e_ie_j+e_je_i = 0$. After a base change in $\mathbb R^n$, the images of the $e_i$ in $GL_n(\mathbb R)$ can be taken to be orthogonal. Then it is immediate to check that the subspace generated by these images is contained in the cone over $O_n$, and has dimension $m$.

Suppose that $\mathbb R^n$ is a module of the Clifford algebra $C_m$, with generators $e_1, \dots, e_m$, and relations $e_i^2 = -1$, and $e_ie_j+e_je_i = 0$. After a base change in $\mathbb R^n$, the images of the $e_i$ in $GL_n(\mathbb R)$ can be taken to be orthogonal. Then it is immediate to check that the subspace generated by these images is contained in the cone over $O_n$, and has dimension $m$.