About the definition of Determinant in Paul R. Halmos' Finite-Dimensional Vector Spaces In §53 he defines determinants:
$\bar{A}$w($x_1$, $x_2$, ..., $x_n$) = w($Ax_1$, $Ax_2$, ..., $Ax_n$), where A is a linear transformation on an n-dimensional vector space V and w is an alternating n-linear form on V. "$\bar{A}w$ is an alternating n-linear form on V, and, in fact, $\bar{A}$ is a linear transformation on the space of such forms". Then he replaces $\bar{A}$ with detA and has (detA)w($x_1$, $x_2$, ..., $x_n$) = w($Ax_1$, $Ax_2$, ..., $Ax_n$).
Does he imply that determinant is a linear transformation on the space of alternating n-linear forms on the n-dimensional vector space V? But I think determinant is an alternating multilinear map over the vector space. What's the meaning of employing the alternating n-linear form w here?
 A: 
Does he imply that determinant is a linear transformation on the space of alternating n-linear forms on the n-dimensional vector space V?

No
In section §53 he says "Observe that det [the determinant] is neither a scalar nor a transformation, but a function that associates a scalar with each linear transformation."  So he is saying det is not a transformation but a function.  It is the function that works as stated: take a linear transformation $A$, create $\bar{A}$ (where $\bar{A}$w($x_1$, $x_2$, ..., $x_n$) = w($Ax_1$, $Ax_2$, ..., $Ax_n$)), and the result is the scalar that scales w by the same amount as $\bar{A}$.
$\bar{A}$w($x_1$, $x_2$, ..., $x_n$) = $\delta$w($x_1$, $x_2$, ..., $x_n$)
This works because w is a (non-zero) alternating n-linear form and the vector space of alternating n-linear forms on an n-dimensional vector space is one-dimensional (see §31).
$\bar{A}$ is a linear transformation on the space of alternating n-linear forms but it is only part of the determinant function as Halmos defines it and not the full thing.

But I think determinant is an alternating multilinear map over the vector space.

This is not the case because an alternating multilinear map takes several vectors as inputs and outputs a scalar.  The determinant, on the other hand, takes one linear transformation as input and outputs a scalar.

What's the meaning of employing the alternating n-linear form w here?

At the end of section §31 Halmos says "The only application [of multilinear algebra] that we shall make is to the theory of determinants (which to be sure, could be treated by more direct but less elegant methods, involving much greater dependence on arbitrary choices of bases)".  So the reason he uses the alternating n-linear form w here is to create an elegant definition of determinants that does not depend on an arbitrary  choice of basis.  A definition of determinants based on a matrix, for example, would require an arbitrary choice of basis because all matrices are defined on a basis. Halmos avoids this.
