I'm familiar with the Young Tableaux method for finding irreducible representations of $\mathrm{SU}(n)$ for $n\in\mathbb{N}$, and I also know of the tensor methods for finding irreducible representations of $\mathrm{SU}(3)$.
Tensor methods for $\mathrm{SU}(3)$ seem to rely heavily on the fact that irreducible representations of this group are identified by two natural numbers $(p,q)$, which define the covariance and contravariance of the tensor associated with that representation. But, if I'm not wrong, and, $n-1$ are needed to identify every irreducible representation of $\mathrm{SU}(n)$. I have no proof of this but it makes sense if we look at the Young Tableaux method.
My questions, then, are:
Do tensor methods for finding irreducible representations of $\mathrm{SU}(n)$ exist for $n > 3$?
If so, which are those?
If not, how is it that they exist for $n \leq 3$, why do they work?