The Grassmannian $$\mathrm{Gr}(2,\mathbb{C}^n) = \frac{\mathrm{SU}(n)}{\mathrm{S}\bigl(\mathrm{U}(2){\times}\mathrm{U}(n{-}2)\bigr)}$$ is the compact Hermitian symmetric space of type AIII of rank $r = \min(2,n{-}2)$.  When $n=3$, this is just $\mathbb{CP}^2$, of rank $1$, so assume $n>3$.  

There is a standard method to classify the orbits of $K$ acting on a compact symmetric space $G/K$ of rank $r$, and you are asking about the special case $K=\mathrm{S}\bigl(\mathrm{U}(2){\times}\mathrm{U}(n{-}2)\bigr)$ and $G=\mathrm{SU}(n)$ and $r=2$.  The basic result is that the space of orbits naturally forms a convex polytope in an 'abelian' subspace $\frak{a}\subset\frak{m}$ where $\frak{g} = \frak{k}\oplus\frak{m}$, and that abelian subspace has (real) dimension $r$.  

In your case, $r=2$, and I *think* I remember that it turns out that the space of orbits is a triangle in $\frak{a}\simeq\mathbb{R}^2$.  One vertex is the fixed point $\mathbb{C}^2 = eK\in G/K$, one vertex is the set of $2$-planes in the $\mathbb{C}^{n-2}$ perpendicular to $\mathbb{C}^2$, and one vertex is the set of $2$-planes that meet each of $\mathbb{C}^2$ and $\mathbb{C}^{n-2}$ in a complex line.  (But don't trust my memory on that.  It could be that one vertex is the set of $2$-planes that are graphs of Hermitian isometries from $\mathbb{C}^2$ to $\mathbb{C}^{n-2}$.  It has been a while since I thought about this.)

A good source for the general theory is O. Loos' $2$-volume work *Symmetric Spaces*.  I would expect this example to be explicitly computed there.