FPS game where skill really matter.

doom

Tribes Ascend , Quake Live , Cube 2 Sauerbraten , Hawken , Battlefield 2142

Halo

Either go for CS:GO or America's Army Proving Grounds. AA:PG is f2p on steam

Definitely CS 1.6

No More Room in Hell

Quake Live.

I'm not sure but...Arma?

You could try Xonotic, an open source quake-like shooter using the DarkPlaces engine. http://www.xonotic.org/

I'm gonna redo things other people have said:

-I know you said no CoD, but CoD 4 is still a blast, at least for me.

• Natural Selection 2. If you play as marines, it has a huge learning curve, and requires incredible skill, and if you get tired of shooting you can play as Aliens. :P

• Left 4 Dead 2. Had a blast with this game, but same as NS2, you don't play a FPS the whole time.

• Heroes and Generals. Free to Play, haven't played it myself, but it looks very interesting.

• CS:GO has been said many times, but I personally didn't care for it. Every match I played there was always some 12 year old kid yelling "GET REKT FAGGOT", or something similar.

• Unreal Tournament. You really can't get any better than this. :P

duck hunt

Fistful of Frags

Planetside 2

Quake III and UT2004.

Counter Strike: Global offensive, no doubt!

Try Doom at harder difficulties. Doom 2 is worth a play too.

like cs ... lol

Wolfenstein Enemy Territory, shame this awesome game is dying, but still better than other fake skilled games based on luck

Duckhunt

Arma series, and to a degree Americas Army too

Planetside 2. If you suck, you will die. A lot.

If you are good you will live a little longer

The difference between first- and third-person shooters is that in third-person shooters the first two persons might get suspicious if they see the gun and stop you from killing the third person. So it's harder.

singleplayer try hard reset and dont die (on hardest diff)
multiplayer if you trust your aim go quake/UT , if you trust your team go cs go go