Dipole nano-antennas have predominantly been investigated in their lateral orientation with their long axes in plane with a supporting substrate. However, the response of coupled dipole antennas oriented vertically to a supporting substrate has so far been out of experimental reach. Here, we present a self-aligned electron-beam lithography technique for fabricating such antennas consisting of metal nanostructures on both sides of a suspended silicon nitride membrane. This 30 nm thick membrane provides an ultra-smooth metal/dielectric interface and uniformly defines the antenna feed-gap size in an array of antennas. It is also a suitable substrate for probing the nano-antenna response with monochromated electron energy-loss spectroscopy (EELS) in a transmission electron microscope. We provide details of this double-sided patterning process, and show the excitation of hybridized plasmon modes in EELS with electrons directed along, and at an angle to, the antenna axis.