Personally, I'd go with Aragorn. It can be argued either way but I based my decision on:
Tolkien states that Legolas accomplished least out of the Fellowship.
Aragorn, whilst not an Elf was trained by them and lived a hard and demanding life as a Ranger.
Aragorn had served both Gondor and Rohan under the alias of Thorongil, before Theoden and Denethor had become rulers. Meaning he probably had extensive combat experience.
But of course, Legolas was skilled certainly.
__________________ Yet the making of things is in my heart from my own making by thee; and the child of little understanding that makes a play of the deeds of his father may do so without without thought of mockery, but because he is the son of his father.
As I stated earlier, Tolkien even said that Legolas accomplished the least of all the Fellowship. He was good, but Gimli beat him in the Battle of Helm's Deep. Aragorn would annihilate him. The only reason people think Legolas is the best is due to the over-hyped movie version.
Legoals killed a fell beast with one shot in the books, that certainly isn't an easy task. Unless you want to dismiss that as luck, the Elf has some serious skills.
__________________ "The most merciful thing in the world, I think, is the inability of the human mind to correlate all its contents. We live on a placid island of ignorance in the midst of black seas of infinity; and it was not meant that we should voyage far." -CoC
Whilst his Elven eyesight and archery skills cannot be questioned, Legolas did possess one of the bows of the Galadhrim.
It is unlikley he could have made such a shot with his Mirkwood bow. Although, I do think that if given the range Legolas would be formidable against Aragorn. Up close, I don't think he's as good.
__________________ Yet the making of things is in my heart from my own making by thee; and the child of little understanding that makes a play of the deeds of his father may do so without without thought of mockery, but because he is the son of his father.
Legolas could just shoot Aragorn in the face and win. There is NO way Aragorn is dodging it. In fact, a lowly goblin archer is theoretically superior to Aragorn.
Another indicator of Aragorns "skill" as it were...or luck is that at the Battle of Pelennor I'm pretty sure it is only Aragorn, Imrahil and Eomer who are listed as having fought through the entire battle and not recieved a single wound.
__________________ Yet the making of things is in my heart from my own making by thee; and the child of little understanding that makes a play of the deeds of his father may do so without without thought of mockery, but because he is the son of his father.
Its a quite a bit different though, being in the midst of a swirling melee of thousands on the battlefield and Mumaks and not taking a single wound, not even a scratch. Tolkien specifically picks out those three for this, implying its important. A lot of other skilled warriors did die at Pelennor notably Halbarad.
__________________ Yet the making of things is in my heart from my own making by thee; and the child of little understanding that makes a play of the deeds of his father may do so without without thought of mockery, but because he is the son of his father.