Keybinding to make End go one character past the last character on the line

By default the End key takes me to the last character on the line in Vim. However, I have set virtualedit=onemore in my vimrc, and I actually want End to take me one character past the last character.

First I tried map <End> g$ but if the line goes off the end of the screen, it only takes me to the last visible character.

Then I tried map <End> $<Right> but if the line is blank it takes me to the next line instead of just doing nothing.

Any ideas?