There are various proof across the site of the equivalence of the Laplace expansion of the determinant and the Leibniz formula for the determinant. There are also various proofs of the fact that the Laplace expansion is independent of the row/column chosen, but they all use induction.
First of all, I take the definition of determinant to be $$ \det(a_{ij})_{n\times n} = \begin{cases} \hfil a_{11} \hfil & \text{if $n = 1$}\\[10pt] \displaystyle\sum_{k=1}^n (-1)^{k+1}a_{1k}\det(A_{1k}) & \text{otherwise}, \end{cases} $$ where $A_{ij}$ denotes the submatrix obtained by deleting row $i$ and column $j$.
It is hard to explain to my students (who have not yet been introduced to induction) why, even though the determinant is defined using the first row, it is in fact independent of the row/column chosen. Is there a proof of this fact which does not use induction or uses permutations? Or at least, a way to illustrate this fact which makes it clear?
Your definition of "det" is itself inductive. If in proving your assertion about $n \times n$ matrices, you're not going to use any properties of "det" on $(n-1) \times (n-1)$ matrices, your definition could equally well be $$ \det(a_{ij})_{n\times n} = \begin{cases} \hfil a_{11} \hfil & \text{if $n = 1$}\\[10pt] \displaystyle\sum_{k=1}^n (-1)^{k+1}a_{1k}\color{red}{\det'}(A_{1k}) & \text{otherwise}, \end{cases} $$ where $\color{red}{\det'}$ is some completely different function (e.g., the zero function).
I think that with an inductive definition like this, you pretty much have to do inductive proofs. (Indeed, your sharpest students might well ask "How do you even know that this defines a function?")
There might be some "way to illustrate this fact which makes it clear", but that really amounts to proof-by-vigorous-assertion.
I guess I'd be inclined to write out the det for a 1x1, 2x2, and 3x3 matrix (with indexed entries rather than numerical ones), and say "Look, all possible n-tuples with one index from each row and one index from each column appear!", and then spend a little while noticing which ones have positive or negative coefficients, and say "there's an alternative definition of det that uses this 'sum of permutations' approach instead...and writing it out is really messy. But let's see, in practice, what it produces if we swap two rows of the matrix..."