I'm sorry, I know this isn't exactly a Mac question but I've tried searching for answers online, and the solutions I found are for Windows. I'm COMPLETELY unfamiliar with computer stuff so if anyone could help I'll be very very grateful. Thank you
Ah, one of my pet peeves with Outlook.com... I'm assuming you're alluding to the Outlook.com web site and not the standalone apps (I don't use MS's standalone apps, including Skype).
MS doesn't allow a user from staying signed out of its messaging service. What I do - and it "sticks" is change my status to "Invisible" to keep the trolls and hookers from hounding me. After setting my status to invisible, I've logged in and out and the status doesn't change. YMMV.