That definitely sounds like a rogue app using the battery more than it should.
Also doesn't even have to be a newly installed app, for instance, I've been using Reddit Sync and lately it's been eating up my battery like crazy, so I think a recent update has caused it.
Either that or my battery is dying.

 

Try looking at the battery statistics. Go to settings(gear)/more/battery. This will show you what is using the most power at the top of the list. You can see if Skype is draining the majority of your battery.

 

Skype was a huge battery hog for me so I quit using it a few months ago. What's even worse is it was hogging all that battery life and I still missed messages all the time until I actually opened the app. It's a horrible app and if you could uninstall it for a day I wouldn't be surprised if it was your problem.

 

If you find you MUST keep Skype on, you can switch to a mega-power battery like the Zero-Lemon 7000mA/hr extended battery. This is what I use. Yes, the battery is big, but your phone will only be 1/2" thick, including the case that comes with the battery. That's not really too big, is it? I carry it around with me in my pocket all day. It can last over 3 days with normal use or 1.5 days with heavy use. You could leave Skype on all day and still have power to spare in the evening. I've had this battery for almost a year and am very satisfied.

Zerolemon Samsung Galaxy S3 7000mAh Extended Battery Black TPU Full Edge Case | eBay