A very popular and easy to use jQuery plugin for fullscreening Google Maps is called gmaps.js. You can find it here: http://hpneo.github.io/gmaps/
Once you've downloaded and included the plugin's JS and CSS files in your project, fullscreening a map is as simple as doing this:
$("#map-canvas").gmap().bind("init", function(evt, map) {
$("#map-canvas").gmap("option", "DROPDOWN_MENU", true);
});
Replace #map-canvas with the ID of your map's DOM element.
And that's it! You should now have a fullscreenable map.