Is there a faster way to swap skills than the fullscreen menu?
Solution 1:
Just about the only thing you can do to speed up the skill switching process is to right click on the skill you'd like to change. By doing this you'll jump directly to the skill selection for that key, bypassing one page of the menus you'd normally click through.
This is the only way to speed up the process of changing skills. There is no other way to automate or hotkey skill selection in Diablo 3. It must be done through this menu.
Any 'out of the box solution' would require, at a minimum, the use of some form of third party macro or automation software, a clear violation of Diablo III's Terms of Use. While such violations are not always detectable or acted on, the possibility of action against your account and revocation of your license to play would give me extreme pause in considering any other solution.
Solution 2:
Currently, there is no faster way to switch the skills. Most likely, this is precisely to prevent what you're trying to do. If Blizzard wanted us to be able to use that many different abilites live during combat, they would've given us more than 6 skills at a time.
On the upside: This makes actual "talent building" and combat preparation for a certain fight pretty important, since you can't rush through the entire game with the same build anymore (at least on higher difficulties), which is a welcome strategic element, in my opinion.
From my perspective it now seems like this is likely something that Blizzard just hasn't focused that much time on, or they weren't able to come up with a better solution. Clearly it would've been possible to achieve the same goals, while keeping the same restrictions and also while making the lifes of players a little more comfortable.
E.g., they could've allowed for Diablo-style skill-switching and just raised the cooldown of newly set abilities that you have to wait for when you want to use them, in order to counterbalance the fact that players are able to switch skills more quickly.