  In this paper we study Steklov eigenvalues for the Lam\'e operator which
arise in the theory of linear elasticity. In this eigenproblem the spectral
parameter appears in a Robin boundary condition, linking the traction and the
displacement. To establish the existence of a countable spectrum for this
problem, we present an extension of Korn's inequality. We also show that a
proposed conforming Galerkin scheme provides convergent approximations to the
true eigenvalues. A standard finite element method is used to conduct numerical
experiments on 2D and 3D domains to support our theoretical findings.
