I am using clipping masks to cut PNG images in illustrator (purely because I don't know how to do it in photoshop) and I was wondering if it was possible to take away the white space that is created from the real image itself.
Yes, it is possible to take away white space from the image itself.
Place an image on the artboard (image format doesn't matter; png's are fine), then follow advice on the images.
When exporting, choose Export for Web and remember to check the Transparency button.
In case you want to use png's with transparency in them in you Illustrator project, just bring the exported image back in and delete the first one you placed on the artboard.
N.B. However, for the purpose of masking images, in most cases that's much more efficiently done in Photoshop.
That's what a clipping mask is for...you'd use that to trace the parts of the image you want visible. However, as far as I know, PNGs don't support clipping paths (I believe you'd need to use EPS files for that).
For that matter, PNGs aren't really a format used much for printing, so I'm wondering if you are instead creating these for the web? If so you'll want to use a transparent PNG which supports alpha transparency. I'm not sure how to export that directly from AI (so would have to defer to someone that knows) but it's pretty trivial to do in Photoshop. You could just cut-and-paste your AI object into PhotoShop then export as a transparent GIF there.