The .ase Adobe Swatch Exchange format is good if you want to open those colors in Photoshop or InDesign. You may have saved this to the default location of \YOUR USER ACCOUNT\Library\Application Support\Adobe\Adobe Illustrator 20\en_US\Swatches. That library folder is hidden, so you will not find that unless you know one of the methods to get to that.
If you want to save the colors from a .ai file to give to someone else you only need to send the .ai file. They can they use swatches flight >> open swatch library >> other library >> and choose the .ai file. they will then get a new palette similar to the swatches palette that has all the colors saved in that document.
You can then select the swatches from the new palette and drag to you swatches if you like.
