Updated Title: Measuring the complete set of spatial Schmidt modes of entangled two-photon fields Updated Abstract: Spontaneous parametric down-conversion (SPDC) is the most widely-used source of high-dimensional entangled two-photon states, and the entanglement in the spatial degree of freedom is considered best suited for harnessing high-dimensional advantages. Although the Schmidt basis provides a natural choice for state characterization of entangled two-photon states in any degree of freedom, the existing techniques can only reconstruct the Schmidt spectrum when the Schmidt basis is known a priori. There is currently no technique that can measure the unknown Schmidt basis of an entangled two-photon state. In this letter, we present a technique that measures the complete set of spatial Schmidt modes without any prior knowledge of the Schmidt basis, as long as the cross-spectral density functions corresponding to individual photons are of Schell-model type. Using this technique, we report measurement of states with over 3000 Schmidt modes–highest reported yet–with up to 98% fidelity. Our technique works even when the entangled field is very low-dimensional or entirely separable, as well as when the field pumping the SPDC is structured. We expect our work to significantly advance harnessing of spatial-entanglement.