In Masoud Kamgarpour's paper "Weil Representations" he uses a set of generators for the symplectic group, referring to a book by R. Steinberg which I do not have access to. If it matters at all, I am working in characteristic zero.
After choosing a symplectic basis, the generators can be written \begin{equation} \left( \begin{array}{cc} A & 0 \newline 0 & (A^t)^{-1} \end{array} \right), \ \left( \begin{array}{cc} I & B \newline 0 & I \end{array} \right), \ \text{and} \ \left( \begin{array}{cc} 0 & I \newline -I & 0 \end{array} \right), \end{equation} where $A$ ranges through invertible matrices and $B$ ranges through symmetric matrices. Does anyone know of a reference or an explanation for this, especially a coordinate-free conceptual and/or geometric one?