Ok, I'm thinking out aloud here so bear with...
Plugging a 120$ BMD decklink mini monitor card into a pc running OBS.
Using OBS screen grabbing functions to grab the Skype window
could then possibly output through the mini monitor. ( SDI or HDMI out )
( I know vMix will def. output over a BMD card, but their 4k
software comes with a cost )
Or
Newtek's NDI connect software can pull in an NDI source
and play it out through BMD cards, So agn a mini recorder card could output Skype's ndi feed.
( As recently introduced Skype is NDI compatible )
As said just thinking out aloud so you might want to follow up on my ramblings.
but either could be a cheap solution