Imagine every program that plays back any kind of sound does that through some kind of output device, like a speaker or headset.
For some programs you can actually choose this device yourself, this includes amongst other programs communication, video and music editing software.
All other software uses the default output device.
With Bandicam you can record what each of these devices output or all of them mixed together.
So to record just the game audio you need to record the device you have selected as default (can be set in the windows sound options - control panel -> sound) and play back Skype through a different device.
In order to do so you have to you have to press Ctrl + , in Skype to open the options and navigate to the sound options where you can select the output device.
Now you just have to record the correct device with Bandicam.
Video Tab->[Recording]->Settings->Sound->Primary sound device
And select your default device (or simply default device which is available on most systems).
Now you should be good.
I hope I was able to help you,
Thank you for reading my post. I hope you understand my "English".
I'm the German translator for some of Bandisoft's products and websites
(mostly Bandicut and Honeycam).