Abstract Magnetic skyrmions, whose shapes are ellipse due to the presence of anisotropic Dzyaloshinskii–Moriya interaction (DMI), have already been discovered in experiments recently. By using micromagnetic simulations, we discuss the ground state and the spin-wave modes of a single elliptical skyrmion in a confined nanodot. It is found that the shapes of skyrmion are stretched into a horizontal ellipse, vertical ellipse, or stripe shape under different strengths of anisotropic DMI. When elliptical skyrmions are excited by in-plane ac magnetic fields, the spin-wave mode contains a counterclockwise rotation mode at high frequencies and a clockwise (CW) rotation mode at low frequencies, and the CW mode depends on the strength of anisotropic DMI. When elliptical skyrmions are excited by out-of-plane ac magnetic fields, the spin-wave mode is split from a simple breathing mode into two complex breathing modes, including a mixed mode of CW rotation and breathing, and another anisotropic breathing mode. Our results provide an understanding of the rich spin-wave modes for skyrmions, which may contribute to the applications in magnonics.