Hausdorff dimension and exact approximation order in ${\mathbb{R}}^n$

Given a non-increasing function $\psi \text{\hspace{0.17em}⁣}:\mathbb{N}\to {\mathbb{R}}^{+}$ such that $s^{\frac{n+1}{n}}\psi (s)$ tends to zero as $s$ goes to infinity, we show that the set of points in ${\mathbb{R}}^n$ that are exactly $\psi$-approximable is non-empty, and we compute its Hausdorff dimension. For $n\ge 2$, this answers questions of Jarník and of Beresnevich, Dickinson, and Velani.