Anti-symmetric representation of $SO(2n)$ to $SU( n)$

in p. 424 of this book, it says that
$$
n(2n-1) text{ anti-sym rep of } SO(2n)$$
$$to
n^2-1 text{ adjoint-rep of } SU( n)$$
$$oplus 1 text{ of } SU( n)$$
$$
oplus n(n-1)/2 text{ anti-sym rep of } SU( n)$$
$$oplus n^*(n^*-1)/2 text{ anti-sym rep of } SU( n)
$$

How do we justify the calculations from eq.12?

I can follow the counntings, but not sure the part of
$$
n otimes_A n^* =?
$$
$$
n^* otimes_A n =?
$$

Text from the book: