For the Stokes problem in a two- or three-dimensional bounded domain with sufficiently smooth boundary, a new mixed formulation is presented by means of a vorticity-velocity-pressure formulation with a new Hilbert space for the vorticity. This new formulation allows arbitrary finite element meshes, especially tetrahedras, in standard numerical simulations based upon the marker-and-cell-method. The appropriate space for the vorticity is a space of square-integrable vector-valued functions with weak rotation that contains the usual space $H(curl, \Omega)$ of vector-valued functions, which, together with their curl, are square-integrable. The new mixed formulation is shown to be well-posed and provides the classical Stokes equations with a new boundary condition for tangential velocity on a subset of the boundary. However, it coincides with the classical boundary conditions for suitable domains such as in the two-dimensional case for a connected, open bounded domain.