I forgot my serial number on my I watch

Couple of things you can do...

1. You can find the serial number in the Watch app at Settings > General > About > Serial Number. You can "long" press on it to copy it to the phone's clipboard if needed.
2. Harder to do/see, but if you take the bands off your Apple Watch you can see the serial number there too.

Here is a support article from Apple with more information...

https://support.apple.com/en-us/HT204520