  Let ${\mathbb V}$ be an $n$-dimensional linear space over an algebraically
closed base field. We provide a classification, up to equivalence, of all of
the bilinear maps $f:{\mathbb V} \times {\mathbb V} \to {\mathbb V}$ such that
$dim(rad(f)) =n-2$. This is equivalent to give a complete classification (up to
isomorphism) of all $n$-dimensional algebras with annihilator of dimension
$n-2$ or, in other words, a classification of the annihilator extensions of all
$2$-dimensional algebras.
