how can I insert an arbitrary sha256 hash into the blockchain?

If I want to prove that I wrote a particular document before a certain point in time, it would be sufficient to generate the sha256 hash of the document and somehow arrange for that hash to appear in the blockchain.

Is there an easy way to do that simply by sending a transaction and leaving it to the miners to arrange for it to get into the blockchain for me?

One way that occurs to me is that I can treat the hash as a private key ( will tell me the address and private key that corresponds to any given input), and send a Bitcoin to the corresponding address. Then when I publish the document anyone can go through the same process and see the date of the transaction in the blockchain.

That seems a little convoluted. Is there a simpler way? Ideally free, not computationally intensive, and not requiring any special software to make the timestamp or to verify it.