An efficient method for the evaluation of the matrix elements of U(2n) spin-dependent generators in a fully spin adapted Gelfand-Tsetlin basis is given. This is done by evaluating the matric elements of the U(2n) generators in a Yamanouchi-Kotani basis whose orbital part is equivalent, up to phase factors, to the Gelfand-Tsetlin basis. This allows the expression for the matrix elements to be separated into products of creation and annihilation operators, which are evaluated using Wick’s theorem, and products of SU(2) Clebsch-Gordan coefficients, whose spin graphs are factorized into easily evaluated segment diagrams. The matrix elements of a single U(2n) generator reduce to a sum of products of segment values. These values are given in formula form involving 3-j and 6-j symbols and in table form, where the formulas have been evaluated for all the nonvanishing segments.