Properties of an alternating bilinear form its coordinate matrix

Hint: Let $\beta$ be a bilinear form and $B$ its coordinate matrix. Applying $\beta$ to two basis vectors $e_i$, $e_j$ gives $$ \beta(e_i, e_j) = e_i^t \cdot B \cdot e_j = B_{ij} $$ that is the $(i,j)$-entry of $B$. Now if $\beta$ is alternating, then it is skew-symmetric, hence $\beta(e_i, e_j) = -\beta(e_j, e_i)$ and $\beta(e_i, e_i) = 0$. Can you relate this to $B$, using the above?

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Edit: As you noted in your comment, this gives $B = -B^t$ with zeros on the diagonal, that is, one direction. For the other direction, note that for arbitrary $x,y$, we have $$ \beta(x,y) = x^t B y = -x^t B^t y = -y^t B x $$ and that $$ \beta(x,x) = \sum_{i=1}^n B_{ii}x_i^2$$