I'm sure I'm not the first person to suggest this but having a free-for-all mode seems pretty standard to me and I can't see a reason not to put it in. Four to Eight people, no teams. Always makes for chaotic fun, may not lean in the competitive direction but offers a lot of potential and variety, also a free for all with no time or kill limit (if the option is available) with a group of your friends is basically just infinite practice/training. It's also a great option for solo playing, when you can't find or are waiting for a team.
I honestly think this would be the biggest bag of chaos one could imagine (not in a good way), where no-one would be able to focus on any fight or any opponent because it would basically be everyone wailing on one another just to do damage. You don't have to worry about friendly fire, so any hit is a point in your favour, and any damage sustained might not be that much of a deterrent, since you can just respawn. There might not even be a reason to lock on to anyone. You could just wade into the fray and spam zone attacks.
Sorry to be so negative, but I'm skeptical of Skirmish as it is, so I honestly don't see this working.