Online skill-based games are online games
If it's about computer games I know that Quake3, Counter-Strike and Starcraft/Warcraft had a fairly big base of serious users. All three are hand-eye coordination type of agility games (plus the last one is a strategic thinking based).
Best Polish counter-strike team has been good enough to win this years WCG (World Cyber Games) in China which is a pretty big achievement. So called proffesional gaming in Poland is still a toddler, I don't know of many polish players beside mentioned counter-strike team who earns for a living by playing games - hence there ain't many polish skill-gaming platforms.
The only one I know is:
And of course particular games have their own main sites with leagues and such..