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Are there “intrinsic” reason why the determinant of a matrix is the product of its diagonal elements?

Let $A \in M_{n\times n}(\mathbb{C})$ be a matrix of size $n\times n$.
I know that the determinant of $A$ is the product of its diagonal elements, i.e. $\det A = \prod_{k=1}^n \lambda_k$. This proof is relatively straightforward if you use the definition of the determinant, but I’m wondering if there is an “intrinsic” way of showing this fact without using the definition of the determinant and a good amount of trigonometry.
By “intrinsic” I mean that the fact that the determinant of $A$ is the product of its diagonal elements is really not a trivial result.
EDIT: for the sake of clarity, I would say that I’m looking for a mathematically rigorous proof, without using any extra formula. In particular, I’m interested in something like the following proof, which is not valid:
Let $V_1, \ldots, V_n \in \mathbb{C}^n$ be some basis of $\mathbb{C}^n$ and consider $A = (a_{kj})_{1\leq k,j\leq n}$ and $d_k = \prod_{j=1}^n a_{kj}$. Show that $d_1,\ldots, d_n$ are the diagonal elements of $A$ and use the fact that $d_k \geq 0 \implies a_{kk} \geq 0$.
Of course, in the case $n=2$ one can simply show that $\det A = a_{11}a_{

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