A linear quadratic model predictive controller (MPC) can be written as a parametric quadratic optimization problem whose solution is a piecewise affine (PWA) map from the state to the optimal input. While this `explicit solution' can offer several orders of magnitude reduction in online evaluation time in some cases, the primary limitation is that the complexity can grow quickly with problem size. In this paper we introduce a new method based on bilevel optimization that allows the direct approximation of the non-convex receding horizon control law. The ability to approximate the control law directly, rather than first approximating a convex cost function leads to simple control laws and tighter approximation errors than previous approaches. Furthermore, stability conditions also based on bilevel optimization are given that are substantially less conservative than existing statements.