  For any right-angled Coxeter group $\Gamma$ on $k$ generators, we construct
proper actions of $\Gamma$ on $\mathrm{O}(p,q+1)$ by right and left
multiplication, and on the Lie algebra $\mathfrak{o}(p,q+1)$ by affine
transformations, for some $p,q\in\mathbb N$ with $p+q+1=k$. As a consequence,
any virtually special group admits proper affine actions on some $\mathbb R^n$:
this includes e.g. surface groups, hyperbolic 3-manifold groups, examples of
word hyperbolic groups of arbitrarily large virtual cohomological dimension,
etc. We also study some examples in cohomological dimension two and four, for
which the dimension of the affine space may be substantially reduced.
