Basis of the space of real skew-symmetric matrices ($\{E_{ij}-E_{ji}:i<j\},\dim =n(n-1)/2$)

For real $n\times n$ skew-symmetric matrices, each upper-triangular entry can be chosen freely, the corresponding lower-triangular entry is its negative, and diagonal entries vanish. Thus the displayed matrix unit differences span the space and are linearly independent: inspecting entry $(i,j)$ with $i<j$ recovers its coefficient. There are $n(n-1)/2$ such pairs, giving the dimension.