  We prove an elementary multilinear identity for the Fourier extension
operator on $\mathbb{R}^n$, generalising to higher dimensions the classical
bilinear extension identity in the plane. In the particular case of the
extension operator associated with the paraboloid, this provides a higher
dimensional extension of a well-known identity of Ozawa and Tsutsumi for
solutions to the free time-dependent Schr\"odinger equation. We conclude with a
similar treatment of more general oscillatory integral operators whose phase
functions collectively satisfy a natural multilinear transversality condition.
The perspective we present has its origins in work of Drury.
