Lieutenant Wolfe is the guy in charge. Sort of.
On paper, he's the boss, but he rarely acts like it. This is partly because of Barnes's superior knowledge and skill set, and the men's loyalty to him, but also partly because Wolfe is weak—plain and simple. He is the prettiest of the guys in the platoon, and this prettiness (the scene where he schmoozes with the guys playing poker captures it best) symbolizes everything: he doesn't fit in, he has no idea what he's doing, and he sure as heck doesn't have the look.
A Man With No Plan
Because he's so weak, Wolfe never really takes a stand for what's right. Ever. When Elias reprimands him for not doing anything about Barnes' execution of the village chief's wife, Wolfe pretends he has no idea what he's talking about. Later, when Elias is talking about the report he plans to file, Wolfe claims he saw nothing. Shortly thereafter, he tells Barnes not to worry about Elias because he, Wolfe, will be on Barnes' side if anything happens.
Wolfe is ultimately killed in the final firefight, and we can see why. The guy is flat out incompetent, a fact nowhere better indicated than in his inability to call in an air strike to the right freaking coordinates (this happens shortly before Elias' death). It makes sense for a guy with no military wherewithal to get killed (O'Neill had predicted it earlier), just as it makes sense for a guy who can't be bothered to take the reins of leadership and do something about the horrors his own men are committing to get killed.