Construction of Grassmann manifolds Is there a way to construct the Grassmann manifold via block matrices? 
For example the upper triangular matrices stabilize the (coordinate) basis of $\mathbb R^n$.  
 A: You can read a construction of the Grassmanianns $G(k,n)$, over an algebraically closed field $\mathbb{K}$, as you required in http://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Hudec.pdf; for exact, the proposition 2.4 at page 4.
UPDATE!
Let $\mathbb{V}$ be a real vector space of dimension $n$, let $k\in\{1,\dots,n-1\}$, let $G(k,n)$ be the set of all $k$-planes of $\mathbb{V}$ and let $\{e_1,\dots,e_n\}$ be a basis of $\mathbb{V}$; defined
\begin{equation}
\forall M\in\mathrm{GL}(n,\mathbb{R}),\,M\cdot\langle e_1,\dots,e_k\rangle=\langle Me_1,\dots,Me_k\rangle\in G(k,n),
\end{equation}
one can consider the action:
\begin{equation}
\alpha:(M,W)\in\mathrm{GL}(n,\mathbb{R})\times G(k,n)\to M\cdot W\in G(k,n).
\end{equation}
One can prove that $\alpha$ is a transitive action, and it turns out that $G(k,n)$ is in bijection with
\begin{equation}
\mathrm{GL}(n,\mathbb{R})_{\displaystyle/P(k)},
\end{equation}
where $P(k)$ is the (closed) subgroup of $\mathrm{GL}(n,\mathbb{R})$ generated by the matrices of the following type:
\begin{equation}
A\in\mathbb{R}_k^k,B\in\mathbb{R}_k^{n-k},C\in\mathbb{R}_{n-k}^{n-k},\begin{pmatrix}
A & B\\
\underline{0}_{n-k}^k & C
\end{pmatrix}.
\end{equation}
For other details, you can consult Warner - Foundations of Differentiable Manifolds and Lie Groups, chapter 3.
