Starting from the classical variational formulation of Stokes problem in mixed form, we propose a new variational formulation. It is obtained with the help of the T-coercivity theory. At the continuous level, the bilinear form appearing in the new formulation is automatically coercive. We then study how to build discrete variational formulations, based on the new formulation. Thanks to the T-coercivity theory, unstable finite element pairs can be stabilized. As a particular case, convergence is recovered for the unstable finite element pair P1-P0. The new numerical scheme is easy to implement. Numerical experiments are carried out for smooth, and low-regularity, solutions of the Stokes problem. We compare the results with computations carried out with the help of the standard nonconforming Crouzeix- Raviart finite element pair P1nc-P0.