The radius and surface composition of an exploding massive star, as well as the explosion energy per unit mass, can be measured using early UV observations of core-collapse supernovae (SNe). We present the first results from a simultaneous GALEX/PTF search for early ultraviolet (UV) emission from SNe. Six SNe II and one Type II superluminous SN (SLSN-II) are clearly detected in the GALEX near-UV (NUV) data. We compare our detection rate with theoretical estimates based on early, shock-cooling UV light curves calculated from models that fit existing Swift and GALEX observations well, combined with volumetric SN rates. We find that our observations are in good agreement with calculated rates assuming that red supergiants (RSGs) explode with fiducial radii of 500 Rȯ, explosion energies of 1051 erg, and ejecta masses of 10 Mȯ. Exploding blue supergiants and Wolf-Rayet stars are poorly constrained. We describe how such observations can be used to derive the progenitor radius, surface composition, and explosion energy per unit mass of such SN events, and we demonstrate why UV observations are critical for such measurements. We use the fiducial RSG parameters to estimate the detection rate of SNe during the shock-cooling phase (<1 day after explosion) for several ground-based surveys (PTF, ZTF, and LSST). We show that the proposed wide-field UV explorer ULTRASAT mission is expected to find >85 SNe per year (∼0.5 SN per deg2), independent of host galaxy extinction, down to an NUV detection limit of 21.5 mag AB. Our pilot GALEX/PTF project thus convincingly demonstrates that a dedicated, systematic SN survey at the NUV band is a compelling method to study how massive stars end their life.