For matrix Lie groups, the exponential map is usually defined as a mapping from the set of $n \times n$ matrices onto itself.
However, sometimes it is useful to have a minimal parametrization of our Lie algebra elements. Therefore, there is the hat operator $\hat{\cdot}$ which maps a $m$-vector onto the corresponding Lie algebra element
$$ \hat{\cdot}: \mathbb R^m \rightarrow g, \quad \hat{\mathbf x} = \sum_{i=0}^m x_i \mathtt G_i, $$
where $\mathtt G_i$ is called the $i$th generator of the matrix Lie algebra $g\subset \mathtt R^{n\times n}$ (especially by physicists). I am very much interested in the properties of this function.
Example for SO3:
$\hat{\mathbf x} = \begin{bmatrix} 0&-x_3& x_2\\ x_3&0,&-x_1\\-x_2&x_1&0\end{bmatrix}$
Question 1: Does the function $\hat{\cdot}$ has a common name? Would "change of basis" transformation (from the standard basis to the Lie algebra basis) be the right name?
I guess in this context it make sense to assume that the family $\mathtt G_1,…,\mathtt G_m$ is linear independent. From the linear independence it follows that $\hat{\cdot}$ is bijective, right? Thus, there is an inverse function $v$ which maps a matrix Lie algebra onto the corresponding Lie algebra:
$$ v(\cdot): g
\rightarrow \mathbb R^m$$
Example for SO3: $v(\mathtt R)= \begin{bmatrix}R_{3,2}\\R_{1,3}\\R_{2,1}\end{bmatrix}=\frac{1}{2} \begin{bmatrix}R_{3,2}-R_{2,3}\\R_{1,3}-R_{3,1}\\R_{2,1}-R_{1,2}\end{bmatrix} = -\begin{bmatrix}R_{2,3}\\R_{3,1}\\R_{1,2}\end{bmatrix} $
Question 2: Is there a closed form solution to write down/calculate the inverse of $\hat{\cdot}$?
I find it a bit confusing that the standard basis vectors are element of $\mathbb R^m$ while $\mathtt G_i$ are element of $\mathbb R^{n^2}$.
(Edit: Heavy changes after I developed a better understanding of the problem.)
(Edit 2: Replaced $[\cdot]$ by the common hat notation $\hat{\cdot}$.)
Best Answer
As to question 1, I'm not sure I'd qualify it as a transform, it seems purely notational to me. Consider the vector space $\mathbb{R}^n$, then the vector $\vec{x} = (x_1\ x_2\ \ldots \ x_n)^T \in \mathbb{R}^n$ is shorthand for
$$\vec{x} = \sum_i x_i \hat{e}_i$$
where $\{\hat{e}_i\}$ is a basis of $\mathbb{R}^n$. Your function, $\hat{\cdot}$, operates in the same manner. In other words, what you think of as your vector is just shorthand.
As to question 2, most likely yes, but it will be basis and Lie algebra dependent.
The confusion you mentioned is really a restatement that unless $m = n^2$, the basis does not span $\mathcal{M}^{n\times n}(\mathbb{R})$ which is okay as $SO(3) \subset \mathcal{M}^{3\times 3}(\mathbb{R})$. Although, if I remember correctly, the basis for $SU(2)$ spans $\mathcal{M}^{2\times 2}(\mathbb{C})$.