Yep! It was apparently tried a few times in Archaic Greece (c. 800-500 BC). This is famously reflected in the Iliad, where Menelaos and Paris try to settle the dispute over Helen between themselves, with the two armies watching (and both sides hoping Paris will die). However, like in the Iliad, the outcome was never respected, and the concept was eventually abandoned. You can read my old answer on champion combat here.