Are there outdoor formaldehyde detection systems available?
What you are looking for is a VOC (volatile organic compounds) monitor or a IAQ (indoor air quality) monitor. They do exist for residential applications and are way much cheaper than the industrial counterparts; however, their detection/sensitivity level may not be as great.

You can try your luck by searching for "iaq monitor" on Amazon and eBay. Prices may vary and be sure to research the product a little, some may only detect CO and other "gases" (LPG, propane, and etc.) and may not make any mention of formaldehyde.

Once you find the right monitor, you will probably have to install on the inside of your home near the window you usually leave open. I don't think these monitors were made to be placed outside.