We introduce 3D optical antennas based on winged nanocones. The antennas support particle plasmon oscillations with current distributions that facilitate transformation of transverse far-field radiation to strong longitudinal local fields near the cone apices. We characterize the optical responses of the antennas by their extinction spectra and by second-harmonic generation microscopy with cylindrical vector beams. The results demonstrate a new 3D polarization-controllable optical antenna for applications in apertureless near-field microscopy, spectroscopy, and plasmonic sensing.