Calculating factorial in terms of certain powers

Question

Factorial $n!$ is defined to be$$n!=1·2·3·4\cdots n$$Let's choose base $2$ to represent $n!$, thus$$n!=2^0·2^1·(2^1+2^0)·2^2·(2^2+2^0)·(2^2+2^1)·(2^2+2^1+2^0)\cdots$$Of course, it is just binary representation of each term of multiplication by entire definition of factorial, but still, is there a gerenzalized formula for calculating factorial for certain natural bases? In particular, for base $2$?