Suppose we have the fundamental representation of $SU(N)$ (represented by a box) which acts on the vector space $V$. Then, the irreducible representation of $SU(N)$ can be found using the rules for drawing young tableaux and symmetrizing/antisymmetrizing indices of the elements in the vector space $V$.
Suppose, $X$ is an element of the adjoint representation of $SU(N)$. To, find the the young tableaux corresponding to the adjoint representation it is assumed that $X \in V \otimes V^*$ where $V^*$ is the vector space on which the conjugate representation acts. It is not clear to me why this is true. Can someone give me some hints as to why elements in the adjoint representation should belong to $V\otimes V^*$?