We present a family of mixed methods for linear elasticity that yield exactly symmetric, but only weakly conforming, stress approximations. The method is presented in both two and three dimensions (on triangular and tetrahedral meshes). The method is efficiently implementable by hybridization. The degrees of freedom of the Lagrange multipliers, which approximate the displacements at the faces, solve a symmetric positive-definite system. The design and analysis of this method is motivated by a new set of unisolvent degrees of freedom for symmetric polynomial matrices. These new degrees of freedom are also used to give a new simple calculation of the dimension of the space of polynomial symmetric matrix fields with vanishing normal traces and zero divergence on a tetrahedron. Such a dimension count was important in the development of the symmetric $H(\mathrm{div})$ conforming methods found in [D. N. Arnold, G. Awanou, and R. Winther, Math. Comp., 77 (2008), pp. 1229–1251].