We can write elements of the $45_a$, where $a$ denotes antisymmetric, as $10 \times 10 $ matrices, because
$$ 10 \otimes 10 = 1_s \oplus 54_s \oplus 45_a$$
Here $10$ denotes the fundamental $10$-dimensional representation of $SO(10)$.
Thus we have for the matrix elements
$$ (45_a)_{ij}=10_i 10_j - 10_j 10_i $$
Why isn't this zero for all $i,j$?