  We study $n$-ary commutative superalgebras and $L_{\infty}$-algebras that
possess a skew-symmetric invariant form, using the derived bracket formalism.
This class of superalgebras includes for instance Lie algebras and their
$n$-ary generalizations, commutative associative and Jordan algebras with an
invariant form. We give a classification of anti-commutative $m$-dimensional
$(m-3)$-ary algebras with an invariant form, and a classification of real
simple $m$-dimensional Lie $(m-3)$-algebras with a positive definite invariant
form up to isometry. Furthermore, we develop the Hodge Theory for
$L_{\infty}$-algebras with a symmetric invariant form, and we describe
quasi-Frobenius structures on skew-symmetric $n$-ary algebras.
