In your statusline, add a %F to display the full path:
:help statusline
" Add full file path to your existing statusline
set statusline+=%F
Note, %F will be the full path. To get a path relative to the working directory, use %f.
If your statusline is not already visible, you may first need to configure it to be always visible, via laststatus=2
set laststatus=2
See :help laststatus for what the options mean. Normally, the statusline may be hidden, or hidden unless multiple buffers are open, but I find it extremely useful to have on all the time with customizations like this, well worth giving up one screen line reserve for it.
The only way I found to get the full path of the file I'm working in is: :echo expand('%:p'). You can re-map ctrl+g if you want, but I personally don't like shifting away from the standards too much. I've mapped F7 like so:
The statusline is very powerful and handy I think. Strait out of the box it will display filename, cursor position and some flags. But you want to do the same as me and replace the filename-part with the full path to the file.
So while editing my .vimrc my statusline could look something like this as default:
.vimrc 26,16 7%
You could view your setting of the statusline with:
:set statusline?
But if you have not made any alterations and no module has changed it it would be empty. But by the examples in the help-section (:help statusline) you could find that the default is:
:set statusline=%<%f\ %h%m%r%=%-14.(%l,%c%V%)\ %P
So include this in your .vimrc and change %f to %F. I also added added the filetype flag (%y) to my statusline since I find it convenient. So my resulting configuration looks like this: