Does this also have a symbolic meaning? In Dutch the translation would be "huid en haar" (very similar), and we use it like when an animal swallows some prey completely, we say "met huid en haar" (med hud og hår). But it is also used in a less physical sense, like when you give yourself completely to a cause we can also use this expression.
If you're familiar with the International Phonetic Alphabet, "hud" would usually be pronounced something like [hʉ:d] and "hode" would be pronounced more like [hʊ:də] or [hudə]. I don't know where you're from, but the distinction between <ʉ> and <u> can be difficult for North American speakers to discern. The web, however, is abound with audio examples =D Lykke til!