Is it possible to design a two player game of skill with absolutely no luck?
By Zermelo's Theorem, every 2 player game has one of the following:


Chance
Hidden information (this also includes simultaneous moves)
Either first or second player has a strategy that will guarantee them the win
Both players can guarantee a draw (or force the game to go on forever)


Any game that has (1) is clearly out. 

Any game that has (2) is also out if you view chance as our way of modeling information that is unknown. After all, the same rules of probability work whether it's unknown because it will be chosen soon by the roll of a die, or is unknown because an opponent has already chosen something but isn't telling us.

Any game that has (3) is out by your definition, which is reasonable because presumably you choose who goes first by chance, and that determines the winner.

So your only choice is to go for (4): games that under perfect play end in draws (or are infinite). For example, tic-tac-toe can be thought of as fitting your requirements because under perfect play every game is always a draw, no chance involved.

Note that we can't always distinguish games in category (3) and (4) - for example, though you talk about the "inherent advantage" of first in chess, it's entirely possible that this advantage is merely an artifact of our incomplete understanding of perfect play, and that two perfect players might always play to a draw (perhaps even in the same way each time). You can also convert any game with (3) into a game with (4) by wrapping two rounds together in a single game where first player alternates, and declaring the overall game a draw unless one player wins both times. So while chess may or may not fit your definition, two-round-chess does.

ADDENDUM, addressing common objections to this answer:


The English definition of "game" is broader and fuzzier than the "extensive-form" definition I am implicitly using to apply Zermelo's Theorem. If you wish to consider other types of games, including timed contests of prowess and trivia, read other answers.
My analysis of (2) is certainly controversial - I consider rock-paper-scissors a game containing luck but you don't have to. For more on different views of chance, read about Bayesian vs frequentist schools of thought.
I consider simultaneous moves under hidden information because you could imagine that players alternate but the first player's move is hidden until the second player moves.