A coherent confocal microscope for fully characterizing the elastic scattering properties of a nanoparticle as a function of wavelength. Using a high numerical aperture lens, two-dimensional scanning and a simple vector beam shaper, the rank-2 polarizability tensor is estimated from a single confocal image. A computationally efficient data processing method is described and numerical simulations show that this algorithm is robust to noise and uncertainty in the focal plane position. The measurement of the polarizability removes the need for a priori assumptions regarding the nanoparticle shape.