  We compute the Hausdorff dimension of the set of $\psi$-exactly approximable
vectors, in the simultaneous case, in dimension strictly larger than $2$ and
for approximating functions $\psi$ with order at infinity less than or equal to
$-2$. Our method relies on the analogous result in dimension $1$, proved by
Yann Bugeaud and Carlos Moreira, and a version of Jarn\'ik's Theorem on fibres.
