Compacting a substitution cipher key
Not quite, if all possible permutations are allowed.

There are $8! = 40320$ permutations over 3 bits; 15 bits of key allows you to specify $2^{15} = 32768$ of them; hence any mapping of 15 bits will necessarily $40320-32768 = 7552$ of permutations unexpressable.

It is doable if you don't allow every single permutation (e.g. allow only even permutations), or if you allow 16 bits of keying data.