So I bought skyrim from target, went to install it on steam, and it says the product key is invalid.

Money well spent, clearly.

Anything I can do to fix this or am I up [censored] creek without a roll of TP?

Check the key closely and make sure you are entering it correct -- Sometimes "O" looks a lot like "0" etc. and you might have made a mistake entering it like missing one digit or entering one twice etc. ! -- If it still does not work call Bethesda's CS and ask them for help - they should be able to verify if the number you are entering is correct or help figure out why it is not working and provide a new number if needed.

There have been roumors of keycodes being stolen and used for pirated games. If you did corectly type in the code you likely fall into this issue.
Beth needs to fix this for you.
Good luck, the game is full of bugs even 11 months after relaese.

If it had been opened and used the error would probably say that the key is already in use not that it is invalid - BUT there was some copies released with the key missing one letter which could be the problem here ! -- Call Beth support and they should help you get it working or give you a new code if it is indeed missing one digit.