This all depends on what you want to focus on. There is no one right way to write a battle scene.
Generally though, it depends on your narration. If you're writing in first person, you'll probably want to focus the most on the things the character sees, so actions and visible emotions (face contorted in anger, focused expression, that kind of thing) - ideally you can combine those by describing how a person moves, for example wide, uncoordinated blows or slow, deliberate movements aiming to probe the opponent.
The surroundings are a bit debatable, but generally, unless you have a very good reason to, don't insert descriptions of surroundings inside a quick, intense battle scene. If you have a bit of a break in the action, you can do a bit longer description of the battlefield, but generally try to describe it together with actions, something like the sword tore the air inches front his nose and embedded into the tree on his right.
In the third person, especially omniscient narrator, you have some more leeway, but the same advice applies. The differences are visible when you have an omniscient narrator, because unlike with first person, you can say a lot of things directly instead of alluding to them through descriptions, mainly emotions.