Do you go to the dungeon?

Conversational initiative

Instead of rolling for initiative and sticking to a turn order, play just flows as part of the normal conversation, just like when you're exploring the jungle or talking to an innkeep. It's not a special "combat mode", you could be talking and then shoot someone and then look inside a chest and then hit someone and then talk some more. It's seamless.

I didn't get to go

Don't worry about missing your turn!

I'm supposed to keep track of actions with cards so that everyone gets the same amount of actions, even if it's sometimes out of order.

But me and the skeleton want to hit each other at the same time

If both of you have actions left that round, see the "First Strike" rules, here. If only one of you have actions left that's the person who gets to hit first.