We consider non-negative solutions of the fast diffusion equation u_t=\Delta u^m with m\in(0,1), in the Euclidean space R^d, d \ge 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t\to\infty for m \ge mc=(d-2)/d, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m \ge mc, or close enough to the extinction time if m < mc. Such results are new in the range m \le mc where previous approaches fail. In the range mc