Sharpness of Rickman's Picard theorem in all dimensions

We show that given $n\ge 3$, $q\ge 1$, and a finite set $\{y_1,\ldots, y_q\}$ in $\mathbb R^n$ there exists a quasiregular mapping $\mathbb R^n \to \mathbb R^n$ omitting exactly points $y_1,\ldots, y_q$.