Let $G\_k(\mathbb{R}^n)$, $n\geq k$ and $G\_k(\mathbb{R}^\infty)$ be the finite-dimensional and infinite-dimensional grassmannians respectively. Their cohomology rings are expressed in terms of universal Stiefel-Whitney classes
$$
H^\*(G\_k(\mathbb{R}^\infty);\mathbb{Z}\_2)=\mathbb{Z}\_2[w\_1,w\_2,\cdots,w\_k],$$
$$
H^\*(G\_k(\mathbb{R}^n);\mathbb{Z}\_2)=\mathbb{Z}\_2[w\_1,w\_2,\cdots,w\_k]/(\bar w\_{n-k+1},\bar w\_{n-k+2}\cdots,\bar w\_n).$$


What are the Steenrod operations $Sq^i$ on these cohomology rings? I want to know the expressions of $Sq^i w\_j$ for all possible $i,j$. 



Steenrod operations on cohomology of grassmannians
As in Prasit's comment, the action of the Steenrod squares on the Stiefel-Whitney classes of any vector bundle are given by Wu's formula
$$
Sq^i(w\_j) = \sum\_{t=0}^i \binom{j+t-i-1}{t} w\_{i-t} w\_{j+t}.
$$
Here $w\_k$ is of course $0$ if $k$ exceeds the dimension of the bundle. Together with the [Cartan formula](https://en.wikipedia.org/wiki/Steenrod_algebra#Axiomatic_characterization), this completely answers your question.


