Which version of Photoshop Elements do you own?
If you have registered the product, you should be able to locate the serial number in your account on Adobe.com.
This link has instructions for the same : Find a Serial Number
Well if you have the first 20 characters and you have access to a Disk Editor, you may be able to search your disk for the first 20 characters and see if you can find the rest of the serial number. I don't know if it works with modern versions of Adobe software, but it used to work eons ago. Of course if you work with a disk editor, you want to make sure you do not edit the contents of your disk by accident. That could be a problem.