This may be the one and only time I recommend staying with your computer. You can run an application like Pamela to record both sides of the Skype call on separate WAV-quality tracks so you can do individual patching and cleanup as needed and then melt the tracks into one show.
http://www.pamela.biz/en/products/
There are other products that semi-do this job, but as you investigate them, remember never do production in MP3 and it’s really good to have split tracks for post production. Also pay attention to which type of computer they work with.
https://forum.audacityteam.org/t/help-recording-with-a-mic-laptop-and-headphones/35068/6
Koz