On the island of Skyros

Skyros did not kill Theseus because Skyros is an island. He was killed on Skyros due to being shoved off a cliff by Lycomedes.

The airport code for Skyros Island National Airport is SKU.

Skyros didn't push Theseus off the cliff. Skyros is a monster, so he forces passerby to kneel to wash his feet. When the person kneels, he kicks him off the cliff into the sea, where his flesh-eating pet turtle is waiting to eat said person. Theseus, on his way to Athens, punished Skyros for his sin by doing exactly what Skyros did to others: Theseus threw Skyros off the cliff where the turtle ate his master. Then Theseus went on his merry way.

skyros killed him by pushing him off a cliff when he was no longer popular in athens.

Edit:

Theseus had fled to the island of Skyros after being declined by his own people because the Athenian's were being told that Theseus was keeping them locked up in the city and took their powers away during Theseus' absence by a noble named, Menestheus.

Theseus died by being pushed off the cliff on the island of Skyros, by the King of Skyros, Lycomedes in fear that Theseus would dethrone him.