The short answer is:
No, you cannot use a derived logo.
From our handy Trademark Guidance page (which you can find in the "legal" link in the footer of every page on our network):
Use of Our Name or Logo in Advertising
- Do use the most current logo where applicable and permitted.
- Do use the proper name for our company, product, or service (see "Proper use of the Stack Exchange and Stack Exchange Name" below).
- Do not use our logo unless you have been granted special permission by Stack Exchange Inc.
- Do not use our name or logo in a way that might imply a false sense of partnership or endorsement with your product.
- Do not use any name or logo that is similar to or might be confused with our name or logo.
- Do not distort or alter our logo in any way.
- Do not use our logo in a way that confuses our brand with another brand.
- Do not use any other artwork from our network without explicit permission.
The good news is:
You may use the actual Stack Overflow logo (available at http://stackexchange.com/about/logos). You should include a disclaimer on the wiki site that says the SO logo and trademarks are used with permission from SE Inc. Please also follow the other trademark and copyright guidelines (e.g., make sure that the page doesn't look like it is own, maintained, or endorsed by Stack Exchange, make sure you properly attribute content taken from SO, etc.).