Thanks to @siggemanen comment linking to this post I learned that the right way to configure how a datatable is serialized when its native WriteXml methods are invoked is to set the ColumnMapping property. In this case to map them as attributes the value is MappingType.Attribute This code shows the behavior

using System;
using System.Data;
using System.IO;
using System.Linq;
using System.Text;
using System.Xml;
                    
public class Program
{
    public static void Main()
    {
        //Get a datatable with 3 columns and two rows for testing purpouses
        var dt = BuildDataTableWithData();

        //To prepare the datatable to serialize columns as xml attributes property ColumnMapping for every column must be set to MappingType.Attribute
        foreach(DataColumn dc in dt.Columns)
            dc.ColumnMapping = MappingType.Attribute;

        //Serializes and display result
        var serialized = Serialize(dt);
        Console.WriteLine(serialized);
    }
    
    private static DataTable BuildDataTableWithData() {
        var dt = new DataTable("table");
        (new DataSet("DataSet")).Tables.Add(dt);
        
        dt.Columns.AddRange(new DataColumn[] {
            new DataColumn("Column_1", typeof(string)),
            new DataColumn("Column_2", typeof(decimal)),
            new DataColumn("Column_3", typeof(DateTime))
        });
        
        dt.Rows.Add("one",1,DateTime.Now);
        dt.Rows.Add("two",1,DateTime.Now);
        dt.AcceptChanges();
        
        return dt;
    }
    
    static string Serialize(DataTable dt) {
        var enc = Encoding.ASCII;
        var mems = new MemoryStream();
        var xmlw = new XmlTextWriter(mems, enc);
                
        dt.WriteXml(xmlw, XmlWriteMode.IgnoreSchema);
        
        var serialized = enc.GetString(mems.ToArray());
        return serialized;
    }
    
}

Pretty simple, you can just use XElement and friends, along with LINQ.

var rootNode =
    new XElement(
        "DataSet",
        dataTable.Rows.Cast<DataRow>().Select(row =>
            new XElement(
                "row",
                dataTable.Columns.Cast<DataColumn>().Select(col =>
                    new XElement(col.ColumnName, row[col.Ordinal])
                )
            )
        )
    );

Console.WriteLine(rootNode.ToString());

dotnetfiddle

If you want to use attributes, just change new XElement(col.ColumnName, ... to

new XAttribute(col.ColumnName, row[col.Ordinal])