Using row reduction, prove that the determinant of a matrix and the determinant of the matrix's transpose are equal.

This is my attempt, but I am not sure if this is actually using row reduction, and if this is the right approach, I am not totally sure on the details:

Suppose $\displaystyle A$ is a square matrix and suppose $\displaystyle E_{1},...,E_{n}$ is a sequence of elementary matrices such that $\displaystyle B = E_{n}...E_{1}A$ is an upper triangle matrix (I am not sure how to prove that such a sequence must always exist). Then clearly $\displaystyle det(B) = det(B^{T})$ since the determinant of an upper triangle matrix and the determinant of a lower triangle matrix is the product of the entries along the main diagonal, which are the same in $\displaystyle B$ and $\displaystyle B^{T}$.

Now this is the part where I am stuck. So I know that $\displaystyle det(E_{n})\cdot\cdot\cdot det(E_{1})det(A) = det(A^{T})det(E_{1}^{T})\cdot\cdot\cdot det(E_{n}^{T})$, but I don't know how to show/prove that for any elementary matrix $\displaystyle E$, $\displaystyle det(E) = det(E^{T})$.

Any help would be appreciated it, thank you.