Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity problem by adding the jump penalty term for the displacement. Here we use the piecewise constant space for stress and the Crouzeix-Raviart element space for displacement. The mixed method is locking-free, i.e., the convergence does not deteriorate in the nearly incompressible or incompressible case. The optimal convergence order is shown in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement, respectively. Finally, some numerical results are given to demonstrate the optimal convergence and stability of the mixed method.