  We generalize the famous weight basis constructions of the finite-dimensional
irreducible representations of $\mathfrak{sl}(n,\mathbb{C})$ obtained by
Gelfand and Tsetlin in 1950. Using combinatorial methods, we construct one such
basis for each finite-dimensional representation of
$\mathfrak{sl}(n,\mathbb{C})$ associated to a given skew Schur function. Our
constructions use diamond-colored distributive lattices of skew-shaped
semistandard tableaux that generalize some classical Gelfand--Tsetlin (GT)
lattices. Our constructions take place within the context of a certain
programmatic study of poset models for semisimple Lie algebra representations
and Weyl group symmetric functions undertaken by the first-named author and
many collaborators. Key aspects of the methodology of that program are
recapitulated here. This work is applied here to extend combinatorial results
about classical GT lattices to our more general lattices; to obtain a new
combinatorial proof of a Zelevinsky--Stembridge generalization of the
Littlewood--Richardson Rule; and to construct new and combinatorially
distinctive weight bases for certain families of irreducible representations of
the orthogonal Lie algebras.
