"Like Quake or Counter-Strike. No like BF or CoD."
.... You need skill in all these games...

Urban Terror

quake, neotokyo, cs in any variant, unreal, shootmania, tf2 (if you're in the correct server)

Verdun!

quake and unreal tournament.

you need hours to learn some moves and stategy.

I know everybody (including yourself, apparently) mentioned it a hundred times already, but...

WELCOME TO QUAKE 3: ARENA

Fistful of Frags

ARMA 2/3

Red Orchestra (the first one). It takes a good amount of time and skill to be able to good. I've played a huge number of the FPS most of people listed here, Red Orchestra is by far the hardest one, you will feel like a sucker for a long moment, it takes time to get good, trust me. But it's a really slow paced game.
If you prefer fast paced game, tough i never played them, the Tribes games are also very popular among skilled fps player.

Try loadout, they told me it's kinda good.
BTW, Titanfall is considered the most competitive fps this year. You shiuld chek it out m8

Fistful of Frags, it's free on Steam and really fun if you're skilled enough :P

Wolfenstein 3D :p

Have you tried Warframe?

Neotokyo, Red Orchestra 2...

Quake Live

Insurgency and ARMA.

Red Orchestra 2: Heroes of Stalingrad
and
Tribes Ascend

• Gotham City Impostors (free to play)

• FEAR Online

• Serious Sam's deathmatch

• Half-life Deathmatch

• Team Fortress 2 (most visible while playing as Sniper, Scout, Soldier)

• Left 4 Dead's versus (16players+)

• Far Cry 3 (multiplayer part)

• Payday 1 and 2

Gunz 2

How about Painkiller (the original, aka "Black Edition", and maybe "Hell & Damnation", aka Painkiller HD)? These are the only "modern"(not really) FPS games that I've played which keep bunny hopping as a mechanic so you have quite the freedom of movement (as well as enemies who will murder your ass on higher difficulties if you don't use it).

While not "skill based" by it's general definition (you can't jump) and without any PvP side, I'd recommend you give Bulletstorm a look. That game gives you quite a lot of options on killing your enemies with a lot of fun factor involved. :D

Call of Duty.

shootmania, fistful of frags, insurgency, hawken

Dino D-Day anyone?
Killing Floor
Counter-Strike series
Quake Live

Not rly skill required but pretty fun: The Ship

Quake and UT, and of course, the lovechild of those 2 - Xonotic!

Rainbow Six: Vegas

or HL(2)DM

Skills uh? Arma 2 and 3, you will have to play for some time until you have a good ratio on this one