Water skiing is available at the Lagoon, either with Eftelya Sport (Apo & co, but a much reduced service from previous years, sadly), and INCI Water Sports, next to Eftelya. You'll have to pay the toll to get into the lagoon for access, or, if you can get their telephone number from Gurkhan (operates the speed boats near to the boat trip section on Belcekiz Beach - Olu Deniz)) they may be able to pick you up from the same boat trip section and you'd avoid the toll.
Alternatively, go to Gemiler Beach, near Kaya, and you could hail one of the speed boats that service the boat trips when they moor up alongside Nicholas Island.
And that's your 3rd alternative - take a boat trip and ski alongside Nicholas Island. But - they want to charge 80TL so try negotiating - we failed to get a reduction this time so didn't bother - I think they are more interested in the Banana rides & suchlike!
Good luck.