Express reflection w.r.t. combination of vectors as combinations of reflection w.r.t. vectors

Question

Let $\alpha\in\mathbb{R}^m$, then for every $x\in\mathbb{R}^m$ we call reflection of $x$ w.r.t. $\alpha$ the reflection of $x$ w.r.t. the hyperplane $\alpha^\perp=\{y\in\mathbb{R}^m\mid <\alpha,y>=0\}$ and we denote it with $\sigma_\alpha(x)$. Then we can compute $\sigma_\alpha(x)$ as $$ \sigma_\alpha(x)=x-2\frac{<x,\alpha>}{|\alpha|^2}\alpha=-\alpha x\alpha^{-1}, $$ where in the last equation we used the Clifford product on $\mathbb{R}^m$. Note that $\sigma_\alpha$ does not depend on $|\alpha|$.
My question is the following: let $\{e_1,...,e_m\}$ denote an orthonormal basis of $\mathbb{R}^m$, is there any convenient way to express $\sigma_\alpha=\sigma_{\alpha_1e_1+...+\alpha_me_m}$ in terms of $\sigma_{e_i}$? Thank you very much.