  An explicit description of the multiplicity space that describes occurrences
of irreducible representations in a splitting of a tensor product of two
irreducible representations of $\mathfrak{gl}_n$ is given. Using this
description an explicit formula for an arbitrary $6j$-symbol for the algebra
$\mathfrak{gl}_n$ is derived. The $6j$-symbol is expressed through a value of a
generalized hypergeometric function.
