conventions - Index of representation of $SU(N)$ fundamental and adjoint

Im getting crazy trying to derive this simple expression.

Say $f^{abc}$ are structure constants of a Lie algebra of $SU(N)$ with $[T^a, T^b]=i f^{abc}T^c$. Then chosing normalization such that

$$\sum_{c,d}f^{acd}f^{bcd}=N\delta^{ab}$$

Show that for fundamental rep of $SU(N)$:

$$tr(T^aT^b)=\frac{1}{2}\delta^{ab}$$

And for adjoint rep:

$$tr(T^aT^b)=N\delta^{ab}$$

Let me know if you have any suggestions, all much appreciated.

EDIT: Adjoint representation case is simply a matter of definition $(T_{adj}^a)^{bc}=-if^{abc}$

$$tr(T^aT^b)=-f^{amn}f^{bnm}=f^{amn}f^{bmn}=N\delta^{ab}$$