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.