It should be 'His face went as gaunt as a skeleton' (your sentence would mean something like 'His face went gaunt after turning into a skeleton'). You can also say 'His face went gaunt, like a skeleton', but it's less correct imo. Generally the way I would describe it is that 'as... as' is used when you're using an adjective that you can turn into a comparative adjective (e.g. tall -> taller). In that case you'd say 'He is as tall as his father' (he could have also been taller), but you'd say 'His car is red, like a Ferrari' (because it's either red or it's not). So in the case of gaunt you'd use 'as' because you can also be less gaunt than a skeleton

(see answer)

Both are fine, but the first sentence sounds more natural.