We consider inverse scattering for the multidimensional Schrödinger equation with smooth compactly supported potential v. We give explicit asymptotic formulas for the Fourier transform Fv(p) at fixed p in terms of the scattering amplitude f at n points at high energies. The precision of these formulas is proportional to n. To our knowledge these formulas are new for n≥ 2, whereas they reduce to the Born formula at high energies for n=1.