Direct or forward wave scattering admits three classical regimes in which the map from scatterer properties or scattering potential to the data is linear, namely, the Born, Rytov, and physical optics approximations. In this paper we derive a new decomposition of the forward scattering map which reveals a previously unknown approximate bilinear forward scattering relation. The latter is data-driven, i.e., it involves exact scattering data, and has the useful property that the dependence on the data and the potential is bilinear. This fundamental result naturally leads to a new linear inverse scattering approach that generalizes and is more broadly applicable than the classical Born-approximation-based imaging. The developed scattering and inverse scattering theory are presented in both plane wave and multipole expansion representations, and the possibility of exploiting support information is also formally addressed in the multipole domain. The paper includes computer simulations illustrating the derived theory and algorithms.