In this paper we present a derivation of the surface Helmholtz decomposition, discuss its relation to the surface Hodge decomposition, and derive a well-posed stream function formulation of a class of surface Stokes problems. The surface gradient, divergence, curl and Laplace operators are defined in terms of the standard differential operators of the ambient Euclidean space. These representations are very convenient for the implementation of numerical methods for surface partial differential equations. Recently we derived a surface Helmholtz decomposition, in terms of these surface differential operators, based on elementary differential calculus. Using this decomposition the variational form of the surface Stokes equation can be reformulated as a well-posed variational formulation of a fourth order equation for the stream function. A particular finite element method for the latter formulation is explained and results of a numerical experiment with this method are presented.