There is a fibration $SU(n) \overset p\to SU(n)/SO(n) \overset j\to BSO(n)$, where the $j$ is the classifying map of $p$, viewed as (the projection of a) principal $SO(n)$-bundle. The Stiefel–Whitney classes for your Wu-esque manifolds are the $j^*$-images of the universal Stiefel–Whitney classes; this is in Borel and Hirzebruch's "Characteristic classes and homogeneous spaces I."
To find a nonzero Stiefel–Whitney number is to find a product of these classes of total degree $\dim SU(n)/SO(n)$. The computation in Mimura–Toda shows that the cohomology over $\mathbb F_2$ is an exterior algebra on one generator each of degrees $2$ through $n$, and that these are the images of the universal Stiefel–Whitney classes other than $w_1$. The product of these generators thus does represent the fundamental class.