  Let $\mathrm{Sym}_q(m)$ be the space of symmetric matrices in
$\mathbb{F}_q^{m\times m}$. A subspace of $\mathrm{Sym}_q(m)$ equipped with the
rank distance is called a symmetric rank-metric code. In this paper we study
the covering properties of symmetric rank-metric codes. First we characterize
symmetric rank-metric codes which are perfect, i.e. that satisfy the equality
in the sphere-packing like bound. We show that, despite the rank-metric case,
there are non trivial perfect codes. Also, we characterize families of codes
which are quasi-perfect.
