Representation of skew-symmetric matrices

Question

It is well known that there is an identification between the Lie Algebra of $3x3$ skew-symmetric matrices and $\mathbb{R}^3$ given by the hat map $$ \hat{ }:\mathbb{R}^3\rightarrow \mathfrak{so}(3),\; \hat{v} = \begin{bmatrix} 0 & -v(3) & v(2) \\ v(3) & 0 & - v(1) \\ -v(2) & v(1) & 0 \end{bmatrix}. $$ My question is: is there some way to generalize this hat map to higher dimensional skew-symmetric matrices? Or, even if not with the hat map, since the dimension of the space of $n\times n$ skew-symmetric matrices is $n(n-1)/2=m$, is there a "standard" way to identify $skew(n)$ and $\mathbb{R}^m$?