  Over fields of characteristic unequal to $2$, we can identify symmetric
matrices with homogeneous polynomials of degree $2$. This allows us to view
symmetric rank-metric codes as living inside the space of such polynomials. In
this paper, we generalize the construction of symmetric Gabidulin codes to
polynomials of degree $d>2$ over field of characteristic $0$ or $>d$. To do so,
we equip the space of homogeneous polynomials of degree $d\geq 2$ with the
metric induced by the essential rank, which is the minimal number of linear
forms needed to express a polynomial. We provide bounds on the minimal distance
and dimension of the essential-rank metric codes we construct and provide an
efficient decoding algorithm. Finally, we show how essential-rank metric codes
can be seen as special instances of rank-metric codes and compare our
construction to known rank-metric codes with the same parameters.
