explicit isomorphism between SO_n defined by two different forms

Question

We know that the special orthogonal group $SO(n,\mathbb{C})$ as a subgroup of $SL(n,\mathbb{C})$ can be represented as a set $\{A\in SL(n,\mathbb{C}): A^tJA=J\}$, for $J$ an invertible symmetric matrix. My question is, how do we set up an explicit isomorphism for two copies of $SO(n,\mathbb{C})$ defined by two different $J$?