I'm reading linear algebra and came up to the topic of linear map representations as matrices. More specifically for a dim $n$ vector space $V$, the space $Lin(V,V)$ of maps $T:V\to V$ is isomorphic to the space of $n\times n$ matrices. This leads to dimension $n^2$.
Let $Mult(V^s,V)$ be the space of all $s$-multilinear maps that send $s$ vectors in $V$ to some vector in $V$. e.g. if $A\in Mult(V_s,V)$, then $A:V^s\to V$ so that $A(v_1,\ldots,v_s)\in V$, where $v_i\in V$.
For $Mult(V^s,V)$, intuitively I can extend the simpler $Lin(V,V)$ case to say that its dimension could be $n^{s+1}$. But I'm not really satisfied.
To explicitly derive the dimension, one needs to come up with a linearly independent that spans the vector space. But I'm struggling to come up with such a set for $Lin(V,V)$ or $Mult(V^s,V)$. Sure I can prove that the former is isomorphic to the space of $n\times n$ matrices, and then go on to say that their dimensions are thus equal. But just out of curiosity, how do I even come up with a basis of linear (for $Lin(V,V)$) or multilinear (for $Mult(V^s,V)$) maps?
Is this even feasible?