The frequencies of spin wave eigenmodes of square permalloy nano-elements with relatively large aspect ratio (thickness 20nm, width 275nm), fabricated by deep ultraviolet lithography, were measured experimentally using Brillouin light scattering (BLS). The measurements were done in a wide range of bias magnetic fields applied along one of the in-plane sides of the square element. The experimental data have been reproduced using both micromagnetic and analytical calculations, which allowed us to reliably identify the spatial structure of the excited spin wave eigenmodes. It is demonstrated that both purely dipolar sinusoidal and dipole-exchange localized spin wave eigenmodes contribute to the formation of the discrete BLS spectrum of the nano-element.