I was sent a link to the video recording made in Skype, and can watch it on the web-based player.
I'd like to download the broadcast to play it on a different device, process it via other tools.
Is that possible at all?
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Sure - there are all sorts of tools to download video when you have a URL.
One that's super easy for me and should be for anyone that has spent a little time learning the bash shell / command - line in the Terminal.app is
Rather than downloading that alone, I would recommend you start with Homebrew package manager:
If you are going to struggle with the command line - this makes it easier to know you've got a good setup, then install the youtube-dl tool (and it's works to download far more then just YouTube, but that's the origin / elephant in the room place where people want to get videos these days).
brew install youtube-dl
Then just copy / paste the URL to see about the Skype download. If you prefer an app you can click, I'd start with the VLC player and open the network location to see how it sees the Skype video.