Let $x^1,x^2,x^3,x^4 $ be the coordinates on $\mathbb{R^4}$ and $p$ a point in $\mathbb{R^4}$. Write down a basis for the vector space $A_3(T_p(\mathbb{R^4}))$.
I have no idea what is the basis for $k$- linear alternating spaces over vector spaces. Can somebody explain it to me. I found some multiindex notaion in a book when looking for basis for space $A_k(V)$, but did not understood it. Please explain.
$\textbf{Notations-}$
Here $ T_p(\mathbb{R^4})$ is tangent space at a point $p$ in $\mathbb{R^n}$ and $A_3( T_p(\mathbb{R^4}))$ means space of all 3-linear function (or also called alternating $3$-tensors, or $3$-covectors) on $ T_p(\mathbb{R^4})$