The author proposes a generalization of the Hermitian and skew-Hermitian splitting iteration for solving positive definite, non-Hermitian linear systems. It is shown that the new scheme can outperform the standard HSS method in some situations and can be used as an effective preconditioner for certain linear systems in saddle point form. Numerical experiments using discretizations of incompressible flow problems demonstrate the effectiveness of the generalized HSS preconditioner.