In the book by Das & Okubo, two different definitions of the adjoint represntation of a Lie group are presented,
For a set of generators {$X_i$},
$(\operatorname{ad} X_i) X_j = [X_i,X_j] = C_{ij}^k X_k $ and
$(\rho_{\operatorname{ad}}(X_i))_{kj} = -C_{ik}^j $,
where $\operatorname{ad}X_i$ and $\rho_{\operatorname{ad}}(X_i)$ are the adjoint representations of $X_i$.
The first definition can be rewritten as,
$(\rho_{\operatorname{ad}}(X_i))_{kj} = C_{ij}^k $, in matrix form.
So, proving the two definitions to be equivalent amounts to showing that $C_{ij}^k$ = $-C_{ik}^j$.
While it is obvious that the structure constants are anti-symmetric in the first two indices, it is not obvious to me why this should be the case for the second and third indices.
Is there a way to explicitly show this, starting with the definition of the Lie product $[X_i,X_j] = C_{ij}^k X_k $ ?
What you are trying to prove cannot be true. Take, for instance, the $2$-dimensional Lie algebra $\mathfrak g$ with a basis $\{e_1,e_2\}$ such that $[e_1,e_2]=e_1$ and $[e_2,e_1]=-e_1$ (of course). Then $C_{12}^1=1$. It would follow from what you are trying to prove that $C_{11}^2=-1$, but clearly $C_{11}^2=0$ (for every Lie algebra, of course, not just for $\mathfrak g$).