Taking for granted you registered your version of Windows 10, you should be able to log in to your Microsoft account and have your license there. If so, you will be able to download it and activate it, but you will have to have some kind of operating system installed in order to get that far, though I do believe it will allow you to download from another computer to a DVD or Flash drive so that you can boot from it and install.
The key is not stored in the BIOS chip. It is just the s/n from the chip that is coded to your account, which is why logging into your microsoft account should unlock and activate your new copy of Windows 10.