Run Skype app. Click on your Profile Picture at the top right corner.
Bigger picture shows up, click on it.
Now Browse and Select the new picture, then Click Open.
I have already tried that, i.e., logging in, and going to my profile page but nothing happens when I "click" (it's a Surface, so a touch screen) on the Profile Picture - which, on my screen is on the left side.
You're welcome – and happy Valentine's day!