Generally speaking mixers used for live sound are too noisy to use for recording applications. So, technically, 'yes' you could hook it up, but realistically, it wont make a very nice noise.
The sound quality would be made even worse if you tried to plug the mixer directly into a mic socket on a PC/laptop as the mic socket will try to amplify the signal (unless it is specifically a line level input - if it's got a picture of a mic, its a mic input, not a line input).
Be careful using USB sound adaptors as they tend to be 16bit rather than 24bit that most audio devices work at - this will further reduce the quality of the signal.
You're best bet would be to get a proper dedicated audio device, you can pick a new one with two mic/line inputs for around £100.