The truth is you've been lied to, or at least that the usual notation makes the innate connection way more difficult to see than necessary.
To see how the two are the same, let me tell you about the wedge product. The wedge product of vectors is like the cross product, in that it is anticommutative--$a \wedge b = -b \wedge a$--but it does not produce a vector. Instead, we directly interpret its result as a planar object--in fact, as the planar object that would be perpendicular to the vector from the cross product.
We formalize that relationship as follows: we say that $a \times b= -i a \wedge b$, where the $i = \hat x \wedge \hat y \wedge \hat z$ is the unit pseudoscalar, representing a volume. The pseudoscalar itself is an object of interest, as it converts wedges to dot products and vice versa when you move it through expressions. In fact,
$$a \cdot (b \times c) = a \cdot (i [b \wedge c]) = i (a \wedge b \wedge c)$$
How does all this wedge stuff connect to linear algebra? Quite simply, actually. Most linear operators will "distribute" over wedges in an intuitive way. That is, for a linear operator $\underline T$,
$$\underline T(a \wedge b) = \underline T(a) \wedge \underline T(b)$$
This is unlike the cross product, which doesn't follow such a simple law. The advantage of this is that, uniquely,
$$\underline T(a \wedge b \wedge c) = \alpha a \wedge b \wedge c$$
for some scalar $\alpha$, for every $a, b, c$. We call $\alpha$ the determinant! It is the special number by which every volume object is dilated or shrunk by the linear transformation, and here it becomes geometrically clear that that is the case: $a \wedge b \wedge c$ is literally multiplied by $\alpha = \det T$ as a result of the transformation.
Now, build up a linear transformation as so. Let $l, m, n$ be vectors, so that a transformation looks like
$$\underline T(a) = (a \cdot e_1) l + (a \cdot e_2) m + (a \cdot e_3) n$$
Then $l,m,n$ are the columns of the matrix representation of $\underline T$, and $\underline T(i) = l \wedge m \wedge n$. This completes the connection between the determinant and the scalar triple product.