We work with symplectic diffeomorphisms of the $n$-annulus${\Bbb{A}}^n=T^*({\Bbb{R}}^n/{\Bbb{Z}}^n)$. Using the variational approach ofAubry and Mather, we are able to give a local description of the stable (andunstable) manifold for a hyperbolic fixed point. We use this in order to geta Melnikov-like formula for exact symplectic twist maps. This formulainvolves an infinite series that could be computed in some specific cases. Weapply our formula to prove the existence of heteroclinic orbits for a familyof twist maps in ${\Bbb{R}}^4$.