ASKING THE IMPORTANT QUESTIONS: Why Does the Bible Have No Recipe for Soap?
Two thoughts come to mind. First, I don't know much about the history of soap, but I kind of doubt that it was in widespread use in ancient Israel or in the ancient world in general. Even in the nineteenth century, Mark Twain had trouble finding it in Europe. (Things are better there now.)
Second, there are lots of things the Bible doesn't include but which must have been well known in ancient Israelite society. For example, the book of Leviticus tells us about sacrificial rites, but includes nothing about the prayers, songs, and music that must have accompanied them and which are alluded to elsewhere (e.g., 1 Chronicles 25).
UPDATE (15 March): I have received a number of responses to this one. For the early history of soap, see here (with the usual caveats about Wikipedia). Also, Gerald Rosenberg points to Jeremiah 2:22, which may refer to a couple of early types of detergent. I suppose the reference to a "fuller" in Mark 9:3 could be relevant as well, since the job involved cleaning wool. But I can't find any evidence for hygienic hand soap as early as the biblical period.