Is your computers IP address public? A whois lookup will give you your routers IP address.
Because Skype and ping use different protocols and ports. It is entirely possible for a computer or router that is up and on the network to refuse to reply to ping requests. Usually non-server computers won't answer a ping, because it tells everyone that the computer is alive and out there, so they can then try other nasty stuff with it (i.e a computer that answers a ping is an invitation to potential hackers). Its also possible that your firewall or router is blocking the ping request or response.
Before you open your computer up to reply to pings, ensure you have secured it against every possible hack out there. Good luck!