Ha ha, there are a lot of people who wish they had your problem :P
Maybe try setting your default communications device also to speakers instead of headset (which I assume is a USB headset?). Then go into your Recording devices > Stereo Mix/What U Hear/Equivalent > Properties > Listen and check the checkbox that says "Listen to this Device", and for "Playback through this device" select your headphones.
So then Skype should output to your speakers, but then also play the sound in your headphones. Then OBS can stream your Speakers, which has both games sounds and Skype.