To move all SKUs to a new custom attribute, you can accomplish that by using product export/import process.
It isn't quite easily to clear the SKU number on each product mainly due to it being use as a product unique identifier. If you want to replace the old SKU with a new one, consider writing scripts that will load product individually by the old SKU, update the SKU to the new one, and then save the product. This process may take awhile to accomplish. (See the relevant question as an example https://stackoverflow.com/questions/28350783/how-to-replace-the-sku-number-for-5000-products-in-magento). This approach will be safer than edit SQL directly since there may be some dependencies to the old SKUs that may end up breaking the system if the reference got altered via SQL DB.
Another option is to export the product to CSV, update the SKU numbers (and move the old SKU to the column for the new custom attribute), then reimport everything into the system.
Hope that helps and good luck!