$SO(n)$'s rep $\rho: SO( n) \to SL(d,\mathbf{R})$ for some dimension d of representations (reps).
The projective group $PSO(n)$'s rep $\rho'$ must be a rep of $SO(n)$'s rep $\rho$. But $SO( n)$'s rep $\rho$ must be a rep of $PSO( n)$'s rep $\rho'$.
But what are the constraints of the $\rho'$ such that which $\rho$ are not survived in $\rho'$?
Does the adjoint rep $(n(n-1)/2)$-dim rep of $SO(n )$ survive in $PSO( n)$?
Does the vector rep ($n$-dim rep) of $SO(n )$ not survive in $PSO( n)$?
What are the rules of the general reps of $SO( n)$ that survived or not in $PSO( n)$?