Hi, Im not 100% sure it's a monitor problem:
1- check in your monitor menu if it's set to auto detect settings, got that kind of problem from an IINC monitor.
2- does your video card have 2 port, you can try the other one
3-check for virus/spyware with malwarebytes
How do you know it's a monitor issue: can you turn it back on or you have to power on you computer once you tried to start skype ? Did you test an other monitor ?