Take a look at the body class tags. This will help you further define your CSS selectors to target the elements you want to hide. In this use case, you can a single CSS rule to hide the header on each and every woocommerce page.
Here is what I would use in your case
.woocommerce header{ display: none; }
There are many different ways to do this. You could use PHP or Javascript to accomplish the same results but I believe this is the simplest solution.