If you ask $p$ questions, then the string of answers is a sequence of $p$ bits, with at most $2^p$ different answers. If the string of answers is supposed to identify a unique permutation, then you need that $2^p\geq n!$ or, equivalently, that $p\geq\log_2n!$.

(The question was updated after this answer and Alon's comment below were written)

If you ask $p$ questions, then the string of answers is a sequence of $p$ bits, with at most $2^p$ different answers. If the string of answers is supposed to identify a unique permutation, then you need that $2^p\geq n!$ or, equivalently, that $p\geq\log_2n!$.