  We study congruences of lines $X_\omega$ defined by a sufficiently general
choice of an alternating 3-form $\omega$ in $n+1$ dimensions, as Fano manifolds
of index $3$ and dimension $n-1$. These congruences include the
$\mathrm{G}_2$-variety for $n=6$ and the variety of reductions of projected
$\mathbb{P}^2 \times \mathbb{P}^2$ for $n=7$.
  We compute the degree of $X_\omega$ as the $n$-th Fine number and study the
Hilbert scheme of these congruences proving that the choice of $\omega$
bijectively corresponds to $X_\omega$ except when $n=5$. The fundamental locus
of the congruence is also studied together with its singular locus: these
varieties include the Coble cubic for $n=8$ and the Peskine variety for $n=9$.
  The residual congruence $Y$ of $X_\omega$ with respect to a general linear
congruence containing $X_\omega$ is analysed in terms of the quadrics
containing the linear span of $X_\omega$. We prove that $Y$ is Cohen-Macaulay
but non-Gorenstein in codimension $4$. We also examine the fundamental locus
$G$ of $Y$ of which we determine the singularities and the irreducible
components.
