I can't fix this for you. But I want you to check and try something.
1. The TV firmware shall be Samsung's latest. I must defer to Samsung on that. Call them to check it out. Ask about Skype too.
2. In the TV change the DNS to 22.214.171.124 and be sure the date and time are correct in the TV and the router. Try again.
Remember Skype is a Microsoft app now. We all know...
My father has a Samsung Smart TV UE40J6300A connected to a BT Hub 5 over Infinity. I've been unable to get Skype to work with this combination.
Skype does work via the same Hub 5/Infinity from both a laptop and an IPad. Skype also works from the same Samsung TV when I connect to the Internet through a different network (via my mobile). Does anyone know if there are any incompatibilities between Samsung Smart TVs and the Hub 5/Infinity please?