even running around all day you wouldn't lose enough oil to harm the engine internally, so you're probably good there, but what I suggest is that you take a lot of pictures of the mess, check the engine over for oil puddles and deal with them, check the oil just to be safe and drive to another garage, if not a few, and act like you left the cap off and get an estimate for repair, then return to the first shop.
Calmly tell the staff what happened and that you want an oil change which needs to include a new filter, any and all fan belts replaced, a new air filter, and a thorough cleaning of the engine and the compartment. This, or a variation of it, is what most mechanics would suggest had you left the cap off claiming they are compromised due to contamination...
check for other items on the estimates and add them too...
hopefully the shop is a descent one and they'll just do it, and not push you into suggesting that they could be looking at an engine replacement because God only knows how truly contaminated the inner engine may be unless they would care to have attorney's decide after reviewing the pictures and the estimates... do not show them these though... they are for your attorney, not them!
good luck!