In this paper we argue for the use of a symmetric bilinear map S on Qn+1 as a means of producing and manipulating symmetric functions; using certain vectors of rational functions we can produce Schur functions. We define S as the determinant of a specific Gram matrix, whose elements are the result of an antisymmetric bilinear map on Q as well as a reversion map R on the same space. Ultimately, S allows us to derive an alternative construction of the Jacobi-Trudi identity (extending the identity to Schur functions) as well as a variant of the Cauchy identities.