Okay, first of all what Operating system vendor (Apple, Microsoft, Etc) are you running? Second, why do you think he's really a hacker and not just some script kiddie?
Take your cable modem offline for about 24 hours so that you pull a new address. If he's still able to get to your machine when the modem comes back online, then he's likely got something on the machine reporting back to him.
If he has software on your machine, he's not going to have messed up your BIOS because, that's ridiculous and you would know for sure if he did something in there and he's not trying to damage your machine because that's how he's tracking you... Also, if he messed up your MBR (Master Boot Record) your computer wouldn't be booting. Oh and tape over any cameras attached to the computer or unplug them if possible!
Also, it is unlikely that it is just Skype if he has actually shut your machine down so, don't assume that.
Once you're sure there's software on your PC, then it is time to contact the authorities but, don't stop with the Police, head straight on over to the FBI site as they're better equipped to handle these intrusions.
https://www.fbi.gov/about-us/investigate/cyber/computer-intrusions
They might want to do forensics on the PC so, don't clean it or reload it, and most of all, don't let the guy know you've contacted the FBI because then he really might mess with the computer (I suggest going to the site from your phone or a work computer).
Anyway, after they do their thing, reload the OS, put anti-malware, and a firewall on the machine (Don't use Windows Firewall if you're running any OS older than Windows 8.
Oh and Absolutely setup all your accounts with 2-Factor Authentication (It will text message you a random computer generated code when you or someone else attempts to login to your account from another computer and they have a valid password and it will prevent them from getting into your accounts).