Find Serial Number of Gateway Without Case

Try running this from a Command Prompt:

H:\> wmic bios list /format

or

H:\> wmic csproduct list /format.

If the serial is kept in the BIOS it should list it with one of these.

Otherwise, it could be on the motherboard, but you would have to physically check.

Source Using Gateway Identity Card to find the serial number:

Using Gateway Identity Card to find the serial number

For computers that shipped with the Microsoft Windows 7 Operating System, Gateway provides an application entitled Identity Card, which can be used to display the serial number for your computer.

To launch Identity Card:

Click Start and then All Programs
Click on the Gateway folder
Click on Identity Card

Alternatively, the serial number will be located in the BIOS which can be entered by pressing the F2 key when powering on the computer.