A general integral formula is analytically derived to represent the wave function of elliptical vortex modes. The derived formula can be straightforwardly employed to take account of spatial damping to explore vortex structure. Spatial damping is found to cause the central degenerate singularity of the circular vortex mode to be split into several singularities with different topological charges. For the non-circular vortex beam, the influence of spatial damping on the distribution of singularities is not significant. Theoretical analyses are confirmed to agree very well with the experimental measurements.