We present a survey of the early evolution of 12 Type IIn supernovae (SNe IIn) at ultraviolet and visible light wavelengths. We use this survey to constrain the geometry of the circumstellar material (CSM) surrounding SN IIn explosions, which may shed light on their progenitor diversity. In order to distinguish between aspherical and spherical CSM, we estimate the blackbody radius temporal evolution of the SNe IIn of our sample, following the method introduced by Soumagnac et al. We find that higher-luminosity objects tend to show evidence for aspherical CSM. Depending on whether this correlation is due to physical reasons or to some selection bias, we derive a lower limit between 35% and 66% for the fraction of SNe IIn showing evidence for aspherical CSM. This result suggests that asphericity of the CSM surrounding SNe IIn is common-consistent with data from resolved images of stars undergoing considerable mass loss. It should be taken into account for more realistic modeling of these events.