We propose a scheme to estimate the parameters bi and cj of the bilinear form z _m = Σ _i,j b _izm ^(i,j) c _j from noisy measurements {y _m } _mM =1, where y _m and z _m are related through an arbitrary likelihood function and z _m ^(i,j) are known. Our scheme is based on generalized approximate message passing (G-AMP): it treats b _i and c _j as random variables and z _m ^(i,j) as an i.i.d. Gaussian 3-way tensor in order to derive a tractable simplification of the sum-product algorithm in the large-system limit. It generalizes previous instances of bilinear G-AMP, such as those that estimate matrices B and C from a noisy measurement of Z = BC, allowing the application of AMP methods to problems such as self-calibration, blind deconvolution, and matrix compressive sensing. Numerical experiments confirm the accuracy and computational efficiency of the proposed approach.